Does anyone know a book about spiders whose bite is deadly, but when people kill them all, they discover that surviving the bite creates immortality?
It may be a short story, I'm not sure.
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Not a spider, but you could be thinking of "Healer" by F Paul Wilson --
"Steven Dalt should have died in that cave on the planet Kwashi. After all, as the natives say, of a thousand people attacked by the cave-dwelling alaret, nine-hundred-and-ninety-nine will die. Dalt survives, but not without personal cost: he has picked up a passenger: an alien intelligence transferred itself from the alaret to take up residence in his brain. Steven Dalt will never be alone again. But Pard, as Dalt names the alien who shares his life, doesn't believe in freeloading. He pays his rent by using cellular-level consciousness to maintain Dalt's body in perfect health" no disease, no aging. And now Dalt appreciates the full meaning of the Kwashi natives saying: Of a thousand struck down by an alaret, nine-hundred-and-ninety-nine will die. But the thousandth will not die . . . ever. Spanning twelve hundred years, HEALER follows Dalt and Pard through the centuries as they become known as The Healer, an enigmatic figure with the power to cure diseases of the mind. And when a wave of interstellar slaughter threatens the civilizations of the LaNague Federation, only The Healer has the resources to face the onrushing doom."
The superhero Spiderman from Marvel comics gains superpowers when a radioactive spider bites him. Not quite immortality, but still quite amazing.
Michael Ende's novel The Neverending Story features Ygramul, a swarm of small flying creatures, that are intelligent and speaking together as a hivemind. The bite of this swarm is deadly in an hour, and that is what it uses to hunt, but the bite also gives the victim the power to teleport himself, which he can use to escape.