Hercules, a.k.a. Heracles, hit his master Chiron the centaur with a poisoned arrow as friendly fire during a battle. Chiron was immortal in the sense that he would never age and would live forever, and also immortal in the sense that he wouldn't die from the poison. Dude was the best healer ever, but he wasn't able to heal himself and the pain was unbearable. Zeus took pity of Chiron and took the centaur's life.

Since this is ancient greek mythology, this would have been contemporary to the story presented in duskwuff's answer.

Hercules, a.k.a. Heracles, hit his master Chiron the centaur with a poisoned arrow as friendly fire during a battle. Chiron was immortal in the sense that he would never age and would live forever, and also immortal in the sense that he wouldn't die from the poison. Dude was the best healer ever, but he wasn't able to heal himself and the pain was unbearable. Zeus took pity of Chiron and took the centaur's life.

Since this is ancient greek mythology, this might have been contemporary to the story presented in duskwuff's answer.