For that matter, so could Kaworu and all the EVAs depending on which interpretation you take.
So does the literal "broken Angel" from the second Rebuild of Evangelion movie which was captured and experimented with until its bones were stripped bare of flesh.
However, actual lore of said Broken "Angels" does not exactly match the trope, since they're really just Sufficiently Advanced Aliens and not Divine Personifications of the Universe. So, Broken Advanced Aliens is closer to their actual definition.
Quite similarly to the examples above, the Organization is Claymore was revealed to have two captured dragon people in their basement, which they all but harvested. The sight of those undoubtedly majestic beings who look like some streamlined science-fiction monsters being chained, hooked up to tubes with half their bodies missing is certainly a textbook example, especially considering how they were being kept just barely alive so their flesh remains fresh for the various uses they have for it...Fortunately Miria grants them a Mercy Kill.
From the fourth arc of Umineko no Naku Koro ni on, a huge theme of the series is whether or not the magic that Beatrice weaves is truly bad or not. Serving to personify this conflict is Beatrice herself, having been turned into an Empty Shell by Battler, who grieves over her until deciding to resurrect her.
Durarara!! has Celty, a dullahan who's lost her disembodied head (along with all the memories in it) some twenty years ago and still feels incomplete without it.
In the first Ghost in the Shell movie, Motoko takes on an enemy one number too big for her and ends up ruining her mechanic body by overexerting herself, leaving her limbs shattered. Helpless, half-naked and unable to move, she makes for quite a sad sight.
While the renegade Shadow Angel Apollonius in Genesis of Aquarion is generally portrayed with his wings spread, one flashback to twelve thousand years ago had him rip his own wings off while escaping from crucifixion to protect Celiane, the love of his life.
Hayao Miyazaki made a music video set to On Your Mark about two police officers who find a winged girl held (and mistreated) by a cult; and then mistreated by the government scientists.
Angel from X-Men has had this happen to him various times in various continuities.
In the original comics, he has his wings cut off by a villain, leading to a period of depression. Apocalypse convinces him to do a Face-Heel Turn by promising to "give them back"... by upgrading him into the angel of death with metal wings. And eventually he manages to heal and regrow his wings naturally.
In the more traditional terms of this trope, we have the Acanti, a race of Space Whale that have been enslaved by the Brood for generations. At least one arc of Uncanny X-Men in the '80s revolved around saving them.
In The Sandman, Lucifer asks Dream to cut off his wings for him since this is the last thing that needs to be done before Lucifer can quit being the ruler of Hell.
He was also Broken already. Probably as a side effect of either rebelling or living in Hell his beautiful white feathered wings had become blackened bat wings.
The Filth gives us Secret Original, a Golden Age superhero who came into reality, and now is a wheelchair-bound cripple.
There are a lot of Good Omens fanfics in which Hell/Heaven catch up with Crowley/Aziraphale and tick them off for the whole averting-the-Apocalypse debacle. In most cases, it's fairly brutal. It often seems to involve their wings, in a hideously painful way.
For some reason, this trope is common in Fan Fiction of Works that include angels/angel-like characters.
At least one of the Alternate Endings to the infamous Cupcakes has this happen to Rainbow Dash. She's saved, but her wings are not.
SWAG.MOV ends the same way: Rainbow Dash ends up destroying Discord, at the expense of her wings.
In fact, "Rainbow Dash loses/hurts her wings" is one of the single most common MLP:FiM fanfic plotlines, right up there with "Twilight messes up a spell".
The Legend of Spyro: A New Dawn ends with Cyros, a dragon who was one of the main antagonists and the first to make a Heel-Face Turn, has her wings horrifically mangled taking an attack meant for Spyro and Cynder. The next chapter reveals they couldn't be saved and have to be cut off. The Spin-Off story detailing her recovery is titled "Broken Wings".
The Ah! My Goddess fic, "Ah! Archfall!" is basically focused on this, when less than perfect human ends up as Master of an Archangel (following the assassination of her Mistress)whilst trying to maintain a relationship with the Valkyrie Lind, who has been banished for failing to recover said angel.
Films — Animation
Toothless from How to Train Your Dragon suffers permanent damage to his tail that hinders his flight capability early in the movie. Hiccup builds a tail attachment to help Toothless fly again, but the dragon still needs a rider to help him fly at his best.
It should be noted that in the short animation Gift Of The Night Fury that continues on from the film, Hiccup builds an automatic tail attachment, allowing Toothless to fly on his own once more. Near the end of the film, Toothless smashes the attachment, resulting in a Heartwarming Moment when you realize that he prefers having Hiccup as a rider to flying on his own.
In Disney Fairies: Secret of the Wings, Tinker Bell has her wings damaged to the point of being unable to fly when crossing over into the country/season of Winter (since her wings can't hold up in cold weather). It's supposed to be impossible to cure, but her twin ends up being a work-around to that.
There's a story told earlier in the film about a Winter fairy that had his wings damaged while courting a Spring fairy. When the Winter fairy telling Periwinkle the story turns away from her at the end, we see that he's been telling his own story; his wings are missing.
In The Last Unicorn Amalthea becomes one after being transformed into a human. Rather than being pleased that her life is no longer in danger (as the antagonist is seeking a unicorn, not a human girl), she's tormented by the feeling that her mortal body is dying around her, cell by cell, and begs to be changed back.
The Hellblazer comic that inspired this event took this even further: after falling from grace and losing most of his wings, Gabriel ends up having the tattered remains chainsawed off by Constantine. And then he's left to make the most of his newfound mortality as a homeless man.
The Angels in Dogma have to lose their wings in order to become mortal.
The destruction of the Plant Elemental in Hellboy II: The Golden Army is treated this way.
Lucifer, the Trope Codifier. As he said in The Prophecy (1995): "I was the first angel, loved once above all others... (sings) But like all true love... one day it withered on the vine..."
The de-horned unicorn in Legend is central to the plot.
Skellig is implied to be an angel exiled to earth, hopeless and miserable but unable to die. In the beginning he lives in an abandoned shed, covered in filth and in such bad shape that he can't fly or even walk, with nothing but insects to eat.
Perhaps a subversion in that his condition (and personality) elicit revulsion rather than sadness and sympathy.
The angel in Date With An Angel gets winged in a collision with a communications satellite.
The felling of the Home Tree in Avatar has a dreadful sense of finality in it and prompts several characters who were hedging their bets to complete their Heel Face Turns.
The dead body of the Kraken in Pirates of the Caribbean is a surefire sign that Beckett is an existential threat to the age of pirates.
In the X-Men: The Last Stand, a young Angel tried to cut his wings off to avoid being rejected by his father.
In the film Noah, the titular character is helped by the Watchers, fallen angels who have been encased in stone for disobeying god by aiding humans, who later betrayed them. They understandably distrust humans.
The titular character of Maleficent after her wings are burned off while she's sleeping. The act is wholly responsible for her start of darkness.
Peter S. Beagle's The Last Unicorn which features the eponymous unicorn being turned into a human girl to save her life. Shmendrick the magician is very happy he managed to do this because his magic worked and he saved her life. Molly, who is a bit more Genre Savvy in terms of unicorns, is horrified:
"What have you done? What have you done?! You've trapped her in a human body! She'll go mad!"
And from the Unicorn herself after she wakes up,
"What have you done to me? I'm a unicorn...I'm a unicorn!"
"But I'm afraid of this human body. More than I was of the red bull...afraid. "
The entire rest of the story could be seen as a rather complex analysis of this trope. The unicorn-turned-human eventually comes to believe that being a human is more like being Cursed with Awesome after falling in love with someone — something that unicorns can't experience. Unfortunately, she still has a quest to finish and abandoning it to live as a human isn't really an option.
Un Senor muy viejo con unas alas enormes (A very very old man with enormous wings) by Gabriel García Márquez is about a strange, injured "angel" who is nursed back to health (kind of, it's a strange short story) and exploited as a freak show by a small town. He has bug infested wings, sounds senile, and is generally decrepit. However, there's no indication that he used to be beautiful. It's just the contrast between what he is and what an angel should be that's sad.
Uncle Einar Has Big Green Wings by Ray Bradbury is a similar story. Einar Elliott normally flies at night so he won't attract attention, but after running into electrical wires, his night vision/radar is damaged, perhaps permanently. His kids come up with an ingenious solution.
In the Magic: The Gathering novels, the angels of Ravnica are the highest authority in the Boros Legion (the law enforcement guild). Then there's the angel Feather, who committed some unspecified crime and got demoted to wojek (essentially, street cop). Notably, she didn't actually have her wings broken off, just bound by enchanted shackles.
In Perdido Street Station, a crippled member of the Winged Humanoid race of garuda comes to New Crobuzon in hopes of regaining the power of flight, before being grounded drives him wholly to despair. Subverted in that, unlike more Woobie-ish examples of this trope, it turns out that his wings were amputated as a just punishment for rape under the laws of his people.
Lampshaded in the Shadowrun novel Lone Wolf, when the protagonist spots a wretched homeless elf dozing in an alley, and wonders what it is about elves that makes such a scene feel far more tragic than seeing a human in similar squalid conditions.
Haunted of Kelley Armstrong's The Otherworld series has Janah, an angel who was driven insane by the Nix.
In L. Jagi Lamplighter's Prospero's Daughter trilogy, the Fair Folk have scars on their back where their wings were amputated after they were shut out of Heaven as fallen angels.
Brom, the aged Dragon Rider from Christopher Paolini's Inheritance Cycle, lost nearly all of his power when his Dragon, Saphira, was killed.
Oromis, also from the Inheritance Cycle, lost nearly all of his physical and magical prowess due to an unknown injury. He is nicknamed "The Cripple Who is Whole," in Broken Angel fashion.
In Skellig, the eponymous character is a weak, arthritic Winged Humanoid found in the back of the protagonist's garage and is heavily implied to be a real angel.
Ray Cummings' Tama Of The Light Country (written in 1930) is about a young Mercutian woman leading a revolt against a hideous law laid down by the wingless males who run the government. Women on Mercury are born with wings — but when they marry, the wings have to come off. As in hacked off without anesthesia. There are several moving descriptions of the victims.
Wheel of Time brings us Stilling/Gentling, (both also known as Severing,) a way to permanently cut a Channeler off from magic. Nynaeve figures out how to cure it, something that was never known even before the last apocalypse.
When Sylphrena's bond with Kaladin is disrupted in Words Of Radiance, we see her losing her mind. It is incredibly disturbing.
Supernatural: In a future timeline, the angel Castiel has lost his angelic powers and operates drugged 24/7 as a cheesy sex guru. He also slowly breaks in the main timeline over season 5, slowly losing his "angel mojo" to the point where he can't heal people and time travel leaves him seriously ill. He also later reveals that he cannot return to Heaven. He does still have his wings, though, and his ability to fly/teleport. At least until the last couple episodes of the season, at which point he can't fly, can't heal himself, and even needs to sleep.
Even though he's been losing his powers, Castiel still held faith in his Father until late in season 5 when Joshua, the one angel who still talks to God, reveals that God doesn't care about the Apocalypse and has no intentions of stopping it or helping anyone, at which point Cas snaps completely. However, at the end of "Swan Song", Cas's angel mojo is restored, with bonuses, when he is resurrected by God.
Castiel still can't catch a break after all of this. In Season 6 he becomes a Well-Intentioned Extremist, loses the trust of his friends and eventually abandons his morals and declares himself the new God. And in Season 7, he goes on a power trip, briefly tries to atone only to get killed from the inside, then returns to life sans memories, regains his memories only to perform a Heroic Sacrifice that results in him going completely insane, and finally gets zapped to Purgatory, where he spends the better part of a year fending off Leviathans. He eventually escapes, sanity and old outfit into the bargain. Now the only problem is Heaven's new management using him as an unwitting spy.
Gabriel also has some aspects of this, namely how jaded he is, and how sick of his family fighting he is. He ran away, only to end up tangled up in the mess all over again, partially thanks to the Winchesters. Being killed by Lucifer probably didn't help.
As of the season eight finale, every angel in the Supernatural universe has been cast from Heaven, apart from the vengeful one who caused it.
Flat-out name-dropped at one point, specifically referencing Cas at the time:
"What are you going to do with a broken angel? Don't be stupid."
In Angel, Illyria finds that her time has passed and her armies are dead. She comments bitterly that in her new body she'd be seen as nothing but food to the rest of her kind.
The Ethros demon from first season also ends up as this—having corrupted and consumed human souls for millennia, it begs for death after trying to possess a child who turned out to be Eviler than Thou.
Many episodes show the Doctor as a broken veteran of the Time War, and the last of his race. In the "The Girl in the Fireplace", Reinette calls him her Lonely Angel.
Torchwood has a hurt, formless creature in the episode "Meat".
In the Masters Of Horror episode "Cigarette Burns", the main character encounters a pale, gaunt, vaguely male figure with healed stumps at its shoulder blades kept by a deranged art collector. It is strongly implied that this was the angel that was mutilated in La fin absolue du monde, the short film that the protagonist seeks. As revealed by the Snuff Film director Dalibor, the sheer sacrilege of this act was the key to the late Bakovic's success and turned his film into a madness-inducing Artifact of Death.
Funny thing, though, is it's a demon, rather than anything divine, as is standard for this trope.
Interestingly, though there are quite a few black angels in Magic which flavor wise are all Fallen Angels, they all have fully functional wings. Only the eponymous Fallen Angel is depicted with broken or amputated wings depending on the artwork.
Werewolf: The Forsaken has a functional equivalent. Spirits known as Lunes are the servitors of Luna, patron goddess of the Forsaken, and provide them with aid and renown. The Bale Hounds, however, have access to arts that allow them to bind a Lune after flaying most of its Corpus away, gaining power from the broken but still "living" wreckage left behind.
Demon The Fallen claims that the entire freaking Creation has gotten this treatment from God during the War of Wrath. The modern World of Darkness has very little to do with the wonders of the young Creation before the Fall, because in Her fury at the rebellious angels, God smashed the world they created under Her supervision and its current state is a long-term result of the damage sustained back then. (The game also offers a counter-perspective, that she was acting to save the world at the Fall - but even so, the damage had been done.)
In Shadowrun, adding cybernetics or modified organs to your body reduces your "Essence" by an amount based on how invasive or alien the part is, and losing essence also causes characters with Magical abilities to decrease their Magic by the same amount (or more, if they only lose part of a point). A magician whose Magic drops to zero due to this becomes "burnt out" - they no longer have access to magical abilities, any magic skills they have become strictly academic knowledge, and they can never regain them aside from extremely rare plot devices or getting a bug spirit shoved into their body (which is at least as bad as it sounds and might well destroy them utterly). The same happens to Technomancers whose Resonance drops to zero.
Exalted has the Demon Realm of Malfeas. Its creation is what happens when you do this to a couple dozen world making titans, turn them inside out, and stuff them inside of one another in a recursive nightmare. Each defeated Primordial was ritually mutilated by having a component soul destroyed. Many demons (who are their remaining component souls) also have a distinct impression of this trope.
Used as metaphor in The Glass Menagerie, with Laura's unicorn figure falling and breaking off its horn.
I'll just imagine he had an operation. The horn was removed to make him feel less - freakish! Now he will feel more at home with the other horses, the ones that don't have horns.
Ico's horns breaking at the end of Ico probably falls under this, though It's open to interpretation. Depending on how you interpret the ending of the prequel, Shadow of the Colossus, the horns could be viewed as the mark of a curse, so Ico's horns breaking could symbolize redemption and the breaking of the curse.
Ein from Riviera: The Promised Land is a literal example, in that he had to lose his angel wings in order to get his weapon. This might be a subversion, though, in that he honestly doesn't care about the fact that he has no wings.
He is rather sensitive about it at the beginning of the game, though, and gets upset when Rose and Marietta point it out.
From the same series, we also have Nessiah from the Yggdra Union games. Because worse things were done to him than his wings and power being taken away, and he's made similar sacrifices voluntarily, not much attention is paid to his physical disfigurements in comparison. He does still seem to be self-conscious about it, though.
Dragon Quest IX's Prologue ends with your hero becoming a Broken Angel, stranded in the mortal realm without their wings and halo and only retaining some of their Celestrial powers. This pales in comparison with some of the other broken angels you meet later.
The hero becomes even more broken near the end of the game when he has to use a Fygg to become a mortal in order to defeat the Big Bad. Fortunately, the post-game gives the hero a chance to regain some of his abilities (ie being able to see Stella and Sterling again) using another Fygg.
After A Taste of Power in the prologue, the Horseman of War in Darksiders is stripped of nearly all of his powers by his superiors after being accused of prematurely triggering the Apocalypse. The rest of the game is spent finding out who is really responsible while rebuilding War's power (and then some since he gets new weapons too). War never regains the ability to maintain Chaos Form indefinitely though.
The elves in the Dragon Age universe were once immortal beings of great magical power, before the humans showed up, caused them to lose their immortality, and demolished their civilization. The elves ran off and made a new home for themselves, and the humans came and destroyed that one too. Today the elves are either enslaved or forced to remain on the run, their civilization regressed to its most primitive, gradually losing their magic, and have lost all knowledge of their history and language.
Both the night elves and blood elves in World of Warcraft have to deal with losing their immortality, because of losing their power sources.
Blood elves never had immortality, the sunwell was a way to curb their mana addictions, but the explosion of the sunwell, disappearance of their prince and subsequent near-genocide of their race did leave them a shadow of their former power.
Trias from Planescape: Torment is an angel whose wings have been reduced to a skeletal husk during his long imprisonment, giving him a somewhat sinister appearance. As it transpired, his wings burned off from his fall. Trias only notes that all the fury of the abyss pales in comparison to the force that did that to him.
In the ending ofKid Icarus: Of Myths and Monsters, Pit's wings fall/rip off in front of a overly happy looking sun, presumably leaving Pit to plummet to his death. No wonder it's called Kid Icarus.
In Kid Icarus: Uprising, Pit's wings get burned up when he insists on flying past the five-minute time limit to save Dark Pit's life. He gets better, though.
NieR has this as one of the major themes, some examples include:
The Shades in general, previously humans who have been reduced to their current shadowy forms; most 'relapse,' losing their minds and eventually die.
Halua having been turned into a human weapon, and then later her brother Emil.
P-33 is a broken death-machine he makes friends with the child Shade Kalil who teaches him to feel and care, until the player kills both of them...
Gretel whose armor/body is mangled by the player and whose companion, Hansel, is killed. Also by the player.
Janus Audron, the Last of His KindWinged Humanoid, who had his heart ripped out by Raziel's past self. Raziel's attempt to revive him is a major plot point. Not that that helped any. Later in the series we see what happens to him after Raziel succeeds in reviving him; he's captured, caged, stuck full of tubes, and drained of blood(as an immortal vampire, this doesn't kill him, only weakens him) to the point where he degenerates into a hideous monstrosity that barely has the strength to stand up.
Raziel himself was once the greatest of Kain's lieutenants. When he revealed his new magnificent wings, Kain jealously though in reality his motives were far more complicated had Raziel executed. Raziel was revived by the Elder God, but he was left a mere shadow of his former self. His once beautiful wings were reduced to tatters, and he himself was a hideous soul sucking wraith.
In Diablo III, Tyrael is a self-inflicted case. He tore off his own wings to become a mortal when he could no longer tolerate his fellow angels' refusal to protect Sanctuary from the demons.
In God of WarAscension, the Hecatonchires is a once mighty precursor to the Olympians who betrayed his blood oath to Zeus and suffered greatly for it. The Furies sealed him in stone, hollowed him out to make a dungeon, and allow Megaera's pet insects to slowly devour him from the inside. He's still alive too.
In Halo, Cortana is an AI that is actually Dr. Catherine Halsey's (*brilliant*) mind. The problem with these AI's is that they absorb more information than they can hold and become "rampant." They start forgetting things and getting mixed up, and going a bit crazy. This becomes a problem for her in the third and fourth games.
In Drowtales, Discodia, Ariel. However technically these are shape shifters so such injuries do not require epic quests.
The muses from Girl Genius, the delicate creations of the greatest spark of the time. Most of them are destroyed or damaged while sparks tried to reverse engineer them. Known examples are Tinka, studied by Tarvek, and Otilia, found beneath Castle Heterodyne. She manages to cause some trouble in her "broken" state though.
Also Castle Heterodyne, an intelligent (and evil, but loyal to the Heterodynes) castle damaged in the Other's attack.
The Severed from Inverloch: Elves who have lost their immortality and ability to use magic. It's revealed at the end that the condition is reversible.
Neopets has the Grey Faerie, a faerie whose wings have been ruined and has lost her powers. Nothing has been done with her aside from her back story sadly.
Though once on their short stories section in their "newspaper", someone wrote a story (maybe it was part of a short series?) that gave her a backstory. Well worth a read if they still do the news paper.
Tan, a character in the RP group Heaven Hell Academy, is a former guardian angel who sacrificed his wings, voice, and eyesight to save the girl he was supposed to be protecting.
Equestria Chronicles has pegasi with torn wings and unicorns with broken horns... who actually seem not to care too much. They'll express their dislike of the situation, but they won't actually act as if they are doomed.
In one episode of Tiny Toon Adventures, Gogo Dodo asks Buster, Babs, and Plucky to rescue his friends from a Circus of Fear. Said friends include a Dragon, a Pegasus, and a Sphynx. The poor creatures are being magically tortured to perform and there's a risk they will disappear forever if they don't return to Wackyland in time.
Played for Laughs in 2 Stupid Dogs. The two unicorns on Noah's Ark get their horns chewed off by the little dog who sees their horns as nothing but tasty bones.
My Little Pony: Friendship Is Magic has Rainbow Dash hurt her wings in "Read It And Weep". She goes stir-crazy with boredom due to being grounded and stuck in a hospital ward. She's still better off than her fellow patient who is stuck in a full body cast. Also happened in the season 2 premiere when Discord took away the wings and horns of the non-Earth Pony Mane Cast members to make his game more "fair". Twilight and Rarity Freak Out.
In Winx Club the Trix destroy the wings of a fairy to distract Musa from their getaway from Alfea's library; they'd set the library on fire, which forces her to save her friend instead of chasing down the bad guys. This bites them in the ass, as the fairy in question was from Musa's homeworld, and rescuing her let Musa acquire her Enchantix form, which conveniently gives her the power to repair her wings instantly.