This trope covers situations when a person is treated as effectively dead and gone even though their body is still physically alive. This is generally because the "person" has been erased in some way.
This can cover mild examples (like memory loss) or more serious examples (like Loss of Identity or damage to the soul). It can occur accidentally or through happenstance (e.g. illness or injury), maliciously (e.g. being assimilated by The Virus) or punitively (Speculative Fiction settings often use mind-wiping and personality overwriting as a form of capital punishment). This trope and its subtropes are often treated as a Fate Worse Than Death or a means of making someone Deader Than Dead. Yet some of the following methods are reversible; an amnesiac could remember everything, a possessing entity could be given the boot, Fake Memories could be recognized as fake. So treat these as "death" only if no such reversal occurs.
Subtropes (examples of which should go on their relevant page) include:
Contrast Un-Person, who is merely treated as if they no longer (or never did) exist. Compare Mind Control and Brainwashed, which usually just force a person to act differently without damaging their personality.
As a Death Trope, all spoilers will be unmarked ahead. Beware.
Touma Kamijou lost his memories in the first arc. The only thing he retained was his Chronic Hero Syndrome. The original Touma returned Index's love note this is more noticeable in the light novels while present Touma is mystified that everyone thinks he would be attracted to a little girl.
The amoral psychopath Teitoku Kakine practically gets erased from existence when his creation, the kind and gentle Beetle 05, takes over his body.
In Letter Bee Gauche Suede lost his "heart" and becomes Noir. He eventually gets better though. Maybe.
This is the ultimate fate of Dream in The Sandman. Dream is one of the Endless, who are practically living ideas, so they can't die in the true sense of the word. After having Mercy Killed his son, Orpheus, Dream gets killed by the Furies, whose job is to punish those who murder their own kin. However, he's reborn in a new form, who then continues to do his precessors work. So the idea called Dream keeps on living, but the character everyone knew as him is dead.
In ElfQuest, this is effectively the fate of One-Eye's body when he is killed in the prelude to The Palace War. After Leetah tries to heal him and only manages to heal his empty, soulless body, it is kept in Preserver webbing for many years until his lifemate, Clearbrook, accepts that he's not coming back and so cuts the webbing and allows his body to die.
The local police chief in the Alex Benedict series was originally a serial killer who had had his memory overwritten. It's a Red Herring, by the way: the apparent Chekhov's Gun never goes off.
The Jokka from M.C.A. Hogarth's Tales Of The Jokka are susceptible to the "mind-death" when they experience physical trauma or heatstroke. Females are most vulnerable, especially during childbirth, but even hardy neuters may fall victim. At the end of A Bloom in the North though they find a Jokka colony on the northern continent where mind death is practically unknown, it's only common on the main continent because of malnutrition in their post-apocalyptic wasteland.
Inversion: In the Discworld series, the Auditors are the supernatural bureaucrats of the Cosmos. They begin as grey soul-less entities; for them, to develop a recognisable personality and individual self-awareness is death. They forgot this in Thief of Time when a group of them decided it would be a good idea to adopt human bodies, so as to tidy creation up from the inside.
In the Chronicles of Chaos by John C. Wright, this is referenced by Vanity, upon learning that all of them are Uranians (or Titans) and prisoners of war of the Olympians, who had their memories erased, forced into human bodies, and raised as such she cries that they commited murder by making them forget their true selves.
In Mockingjay Katniss feels this way about the hijacked Peeta. She grieves him as if he had died.
The Demolished Man provides the inspiration and perhaps Ur Example for the trope namer with the practice of Demolition: erasing a convict's mind to a Blank Slate and building it up again into a more benevolent personality, thus not removing an intelligence that could possibly be beneficial to mankind just because the personality was bad.
In the episode "The Quality of Mercy" a serial killer is sentenced to Death of Personality, because he's considered too dangerous to ship back to Earth and military law only allows spacing in cases of mutiny and treason.
The punishment is deconstructed in the episode "Passing Through Gethsemane" when one of an order of monks living on the station is horrified to find out he's actually a convicted serial killer who was mind wiped and reprogrammed with a personality inclined to do service. When his detention center had a fire he got lost, and eventually found his way to the monks.
An episode of Law & Order: Special Victims Unit ends with Munch's clinically depressed uncle coming off his medication to "kill" himself as penance for murdering a suspect while in a mania caused as a side effect of his medication (the medication he was offered was explicitly stated as avoiding these side effects again).
In the Buffyverse, vampires in general are portrayed as this. They're essentially a human who's died and had their soul replaced by a demon (who gains their memories and sometimes might even actually believe they're the original person). The exact implications of this aren't clear (vampires have been shown to be able to regain a soul), but this trope has been brought up in regards to it several times. The best being the episode "Lie to Me", where Buffy tries to explain this to a group of Vampire Vannabes (and the episode's villain, who it turns out doesn't care, because he's dying anyway).
Fred had her soul completely destroyed when Illyria possessed her in the final season of Angel. People, including Buffy herself, came Back from the Dead several times, but it was made very clear that Fred was Killed Off for Real, even though her body was still walking around and could do a very good impersonation of her.
People who are transformed into Cybermen are stripped of all personality and individuality, becoming soulless killers. Once a human is transformed, they're considered dead and all that can be done is to destroy the Cyberman.
Everlasting death for the most faithful companion — loss of memories causing the reversion of a significant amount of character development.
In the Star Trek: Deep Space Nine episode "Sons of Mogh" this ends up being Worf's solution for dealing with his suicidal brother Kurn, who had demanded that Worf kill him so he could die with honor. He hit on this solution after being rather unsurprisingly forbidden to carry out the normal ritual for this. One memory erasure and plastic surgery later, Worf hands Kurn off to a friend of their father's who had promised to treat Kurn as his own son.
Shadowrun. When an insect spirit possesses a human being it overwrites and destroys the human's personality with its own. If the spirit achieves a "Good Merge" it can keep the original personality's memories and use them to impersonate the victim.
"If you become a bird, your mind becomes a bird's. Isn't that the same as dying?"
Inverted in Final Fantasy VIII, where the entire cast is victims of some form of dead personality ( the SeeD are amnesiacs, the sorceresses, Edea and Rinoa, are victims of possession by Ultimecia), but at the end of Disc III, everyone gets better.
In Freefall this is the result of the "Gardener in the Dark" upgrade for robots using Bowman's neural nets. It cuts off all neural pathways not related to their core programming, a Jar Jar Binks robot that was determined to scrap himself and quoting Shakespeare started running around saying "meesa Jar Jar Binks" when infected, and falling over and requesting assistance getting back up. Fortunately it can be canceled if the robot hasn't had a chance to sleep and commit the upgrade to long-term memory, just wipe their day memory and they're back to how they were the previous day.
Happens in WALL•E after Eve repairs him from near-fatal damage. He appears to lose all his memory and personality, until she "kisses" him with an electric spark.
In Brave, Merida and Elinor must Race Against the Clock to break the spell that gave Elinor a bear's body or she will lose her humanity and become like any other non-sentient bear.