The first season of Pokémon used this trope quite a lot, such as when Ash first meets Charmander and when Pikachu needs medicinal attention after getting mobbed by Spearows in the first episode. As the seasons went by, depictions of pokémon sustaining very serious injuries became less common, causing this trope to be used less frequently than it once did.
In Legion of Super Heroes, Mon-el, Ultra Boy, and the White Witch found Dev-em prisoner and dying of kryptonite poisoning. They swept in to seize him and bolted off with him — Ultra Boy nearly panicking because he could not find a pulse — and back to their starship, where Mon-el set the machinery to cure him.
In a "What If" story, Doctor Doom mind-controlled Mr. Fantastic to get him to attack the Fantastic Four; though he broke free, he was badly injured. His lab assistant Lysette kept him alive while urging them to fly the faster to the hospital.
When Rogue had just joined the X-Men, she was mortally wounded fighting next to Wolverine, saving his fiancee. Wolverine compelled her to absorb his healing powers to save her life.
The "Fatal Attractions" X-Men crossover climaxes in a confrontation with Magneto, where Magneto rips out the adamantium bonded to Wolverine's skeleton. The final issue (Wolverine #75) is concerned primarily with the team trying to keep Wolverine alive and stabilize him. The trauma was so severe that Wolverine's Healing Factor effectively "gave out" after he was stable, and took months to start working again.
In I Did Not Want To Die, this trope is subverted because it's a mortal wound and it occurs in a break in the action.
"Iron Star" is all about Curth and Dorden's fight to save Gaunt — told almost entirely from Gaunt's point of view, meaning he has only minimal awareness of reality in the midst of a dream.
In Salvation's Reach, Dorden is afraid they are unable to save Cant's life, because removing the garotte will release the pressure that was holding the blood in the artery. Kolding thinks of a means, and they labor for hours, succeeding.
In J. R. R. Tolkien's Fellowship of the Ring, Frodo has to be rushed to Rivendell after an injury sustained at Weathertop threatens to turn him into a wraith if he's not treated soon and properly at Elrond's house.
Late in Deryni Rising, Morgan is injured while fighting a traditional sword duel as King's Champion, defending Kelson's right to be crowned King Of Gwynedd. While Kelson personally fights an arcane duel against Charissa to resolve the issue, Alaric asks Duncan to help him Heal himself. It's only the second time Morgan has ever done it, and he's lost some blood, while Duncan has never even tried to Heal before and doesn't know if he can do it at all.
In The Hunger Games when Katniss finds Peeta in the arena he has been stabbed badly by Cato and lying in by a river camouflaged with mud hasn't done wonders to prevent infection. She does what she can to help him but it's obvious to both of them that without medicine he's not going to make it. So she risks herself to get the medicine.
In James Swallow's Warhammer 40,000 novel Hammer & Anvil, Verity cites the possibility of this as a reason to keep her near the scenes of danger.
In Sarah A. Hoyt's Darkship Thieves, Thena rushes Kit off to Doctor Bartholemeu after Joseph tries to kill him. Given that she doesn't know the way there, and Kit can only communicate it telepathically, it gets difficult.
Bleeding and weak I reached my women, who, accustomed to such happenings, dressed my wounds, applying the wonderful healing and remedial agents which make only the most instantaneous of death blows fatal. Give a Martian woman a chance and death must take a back seat. They soon had me patched up so that, except for weakness from loss of blood and a little soreness around the wound, I suffered no great distress from this thrust which, under earthly treatment, undoubtedly would have put me flat on my back for days.
A Local Habitation, after Toby just barely gets Quentin out of the way of being shot and killed, he still takes a bullet and they have to rush for this. Later, when Toby confronts Gordan, she has to worry about Eliot bleeding out in the meanwhile. Subverted there; April bound his wounds during the fight.
Ashes of Honor, what appears to be an After-Action Patchup transforms when Tybalt collapses. Toby has to rush to arrange for the healer to be brought in — and allowed in by the magic.
In Patricia A. McKillip's The Riddle-Master of Hed, the voiceless Morgan is attacked on the plain by apparent merchants; Astrin hauls him back to his cottage, threatens off another merchant, and brings in an old woman to treat his serious wounds.
Averted in The Leonard Regime. Brandon at first appears to have taken a critical hit, but it later turns out it only grazed his skin and he passed out from fright.
In Julie Kagawa's The Iron Daughter, Ash is seriously wounded before Meghan works out how to remove Virus's bug and stop its mind control. Puck insists on getting him to the healer.
Downplayed in The Lion, the Witch and the Wardrobe: Edmund lies gravely wounded after the climactic battle, but Lucy saves him with her healing cordial quickly and with a minimum of angst. Played straighter in the 2005 film version, which drags the scene out a bit longer and gives him a brief Disney Death.
In Charles de Lint's Seven Wild Sisters, when Sarah Jane first sees one of The Fair Folk, she has to pull out the poisoned arrows he was shot with, and then bring him to Aunt Lillian for more treatment.
In Richard Ellis Preston Jr.'s Romulus Buckle & the Engines of War, after their encounter with the sabretooth beasties, Romulus has to carry Max off, find shelter, and treat her injuries.
Live Action TV
This is essentially the point of a Mash Unit - get the wounded to a doctor who can patch him up as quickly as possible. And they get the patients from Battalion Aid, which is even closer to the front lines.
Merlin: In 'Le Morte D'Arthur', Merlin must race to find a solution for Arthur's mortal injury from the Questing Beast, which can't be cured by normal means. And there's 'The Poisoned Chalice', where Arthur races to find an antidote for the poisoned Merlin. Much later, there's a try-and-fail with Arthur and Merlin trying to save the dying Uther in "The Wicked Day".
CSI NY "Page Turner" the gang had to race to find out exactly what type of radioactive substance had caused two people to die and the coroner, Sid, to collapse, so they could tell the doctors what to do to treat him. Mac's "Near Death' ep may or may not count...I'm not sure because the doctors were racing to save Mac while the team raced to find the shooter.
In the first episode of the second season of Breaking Bad, after Tuco has delivered a No-Holds-Barred Beatdown to an unwise subordinate, said subordinate gets dragged into a car to sleep it off. A moment later, the car comes roaring back up to Walt and Jesse with Tuco jumping out and demanding that Walt do something, "because you're smart!" as the subordinate is in bloody convulsions. Walt tries some CPR, only for reality to ensue.
Angel, "A Hole In The World", where the team races to save Fred after Illyria infects her, ultimately failing.
In Rudyard Kipling's "The Married Man", contrasting the fighting of the bachelor and the married man, he observes that the married man is best for saving your life in this.
Deadlands had these scenes virtually built in to the ruleset: a lasting injury—however minor—could only be conventionally treated in the "golden hour" after it occurred, often leading to a mad scramble after a fight. It was also absurdly easy to bleed out—a Time-Delayed Death, but with more fainting—virtually guaranteeing that at some point, one member of the posse would have to drop out of combat to patch up one of their teammates.
In World of Warcraft, the Pandaren starting zone is a giant turtle island. Near the end, you have to remove a ship that has crashed into and pierced its flesh... by blowing it up, which leaves a huge bleeding wound. The next quest is to protect a bunch of healers who are trying to heal the wound before it dies and sinks.
Gil collapses after he stands off the attackers, and the jaegers get him to Mama Gkika's. This at first appears to be just sneaking about, but there we learn that Mama resorted to some heavy duty medicine to save and revive him.