Airman Higgs got Baron Wulfenbach out of a burning airship and to safety. During which, he had to deal with a bunch of monsters, an irate Captain Dupree, and a goose, breaking three out of four limbs in the process and getting an infected bite. He managed to get to a town despite his injuries, where he got shot in his last undamaged limb. He informed the local garrison of Wulfenbach's predicament, then passed out.
Then shows up here and here fighting off a Muse that is possessed/occupied by the Heterodyne Castle's AI.
The canadian lumberjack in Antihero for Hire certainly adheres to the trope strongly, as seen here. Some of MANTIS's agents, especially the elites, qualify.
Fern from Awful Hospital, fighting her way through incomprehensible weirdness to get her kid back.
Jozk of Charby the Vampirate refuses to fail his mission, fighting someone he knows is an immortal teleporter, managing to actually delay her even after she gives him fatal injuries, and continuing on his way until his body literally cannot move anymore.
Laura from Collar 6...well, when Ginger is watching, anyway.
Ludwig in A Day With Bowser Jr. He is so hell-bent on murdering Bowser Jr and claiming back the Koopa throne, that literally all of his dialogue revolves around his desire for the throne. He's even thinking about killing Bowser Jr while bedridden and severely injured!
Denma The Quanx: Edel. In his pursuit of his goal, which is to live freely with his True Love Nell he: spends time in prison, loses over 100 pounds, becomes a guard, leaves the church, sells his body as a prostitute, kills a dozen people, escapes prison (twice), walks directly into a war zone, and steals a spaceship.
Dominic Deegan has The Infernomancer. No matter if he's devestated by holy magic, blasted by chaos infused black magic, banished to an alternate dimension full of eldritch abominations, or sent directly to hell, he just... Doesn't... Die...
On a more humorous note, Dabura manages to survive a giant fucking hole in his chest far longer than should be possible.
The same can be said of U13 Kakarot who survived a sucking puncture wound long enough to wake up and call for medical attention, when everyone fully expected him to be dead after U18 Vegeta was through with him.
You will do it. It might work, it might not, but you will do it. And you will keep doing it, for her. Because you are relentless. You will never stop doing whatever you can for her, ever. And because of that, ultimately, she will be fine.
El Goonish Shive: Nanase was the only one able to protect Ellen from the Knight Templar trying to kill her, but the only thing she could do was summon a doll-sized version of herself at Ellen's position, though she could still throw full-powered punches in that form. She would get a few hits in, and then the Knight Templar would destroy the doll, which would make her experience the pain of death. So she'd summon another one...and another....and another...
Sollux does it again when he telekinetically flies the troll's asteroid toward the beacon of the Green Sun, their only hope of survival, at the rapid speed necessary to get there before the beacon's path fades. The strain is so great that he bleeds from every orifice before ultimately dying permanently of it.
Terezi Pyrope also deserves mention, having raised herself almost completely alone from infancy on Alternia, a freaking Death World.
This would be amazing already without the fact that she is blind.
Kanaya after getting fatally shot by Eridan, manages to pick herself up and escape offscreen. During Nepeta's time wandering around she finds a trail of Jade blood and a destroyed transportalizer. When she confronts the three who've gone rogue she's Eridan's cape around her wound as a sash, which has since soaked through with her blood and she still takes them down.
GamzeeMakara has recently taken things to a new level with the revelation that he has managed to survive through will alone, throughout every timeline, including ones in which a psychic shriek was emitted that is fatal to his entire species. He's also still moving in the alpha timeline, despite taking 5 minutes worth of automatic fire from Caliborn's Machine Gun.
Vriska is obsessed with being the most important character of the plot, essentially orchestrating (Or at least so she thinks) every major event of the Alpha timeline. Even after she died, she's still hell bent in being the one to defeat lord English by rallying up an army of ghosts in the void between universes. Taking this trope to its logical extent, she will even directly challenge the narrator/writer of the comic psychically in order to get back in the spotlight and finish her Story Arc properly.
Perhaps more impressive are her actions during the revenge cycle. After Terezi causes the cueball to blow up in her face (removing her right arm and eye) she regains consciousness and immediately performs a double psychic reacharound (mindcontrolling Tavros to use his powers to control a dragon to use its connection to Terezi to mind control her into blinding herself) to get her revenge, and is still mad enough to taunt Terezi over a chat client before passing out due to bloodloss.
Karkat might qualify as well, because on Alternia, being (as previously stated) is a Death World, where just surviving is a feat, not only does he survive, he does so living in a world where his ENTIRE EXISTENCE is a crime punishable by death because of his blood colour, but he does so while living in close proximity to other trolls. Which means, because of troll biology, he would never be able to cry, blush, bleed or have sex. EVER. The last of which, in troll society, also means death. Terezi teases him about it mercilessly.
Caliborn. It is impossible to win a session with just one player. Caliborn does it anyway. He is a Time player, which means he can go back in time and redo things until he gets them right. Out of sheer stubborness, he achieves all of the levels, by himself, just by level grinding for years and years, solves all of the puzzles despite his learning disability, and eventually becomes the most powerful villain in the universe. On a lesser level, Caliborn's art is absolutely terrible. Everyone who sees it tells him that it is, and that he should stop drawing, but he draws anyway, confident that he will become a great artist. His art has improved immensely from the start of the story, from "incomprehensible" to "bad" to merely "poor", so that just goes to show he was right not to listen to the haters.
When Sammy arrives at the eponymous show in The Sanity Circus to find Attley, he will not let anything stop him. Not having to tear down the circus, not even being thrown head-first into a brick wall by magic. It takes a Safeguarde to stop him, and even that's because he was close to being Impaled with Extreme Prejudice.
Ever since his My Greatest Failure, Lalli would rather repair a mistake he did in a task while half asleep rather than leave it hanging. As one can guess, this tends to make him blind any new mistakes he may be making in the process until their consequences actually halt his progress.
The quality is greatly associated with Swedes, both by themselves and other nations. This ranges from maintaining a railway system in a country that consists of various settlements separated by Death World areas to Emil's own persistence at befriending a No Social Skills mage with an Ambiguous Disorder who speaks a language completely different from his own.
Anne from Sunstone puts a startlingly large amount of time and effort into getting Alan to cross dress. This infliction was caught by the fans when Sejic hinted this would actually happen after an outrageous amount of page views.