There are several other personages known as chiranjivins, like Jambavantha (Jambavan). However, in Hinduism, "immortal" does not mean eternal. Even immortal things are dissolved at the end of the universe, including its secondary creator Brahma. The only eternals are Vishnu and Shiva of the Trimurti, i.e. manifestations of Brahman (the Supreme Reality), Sheshanaga , and the four Vedas.
At the end of one universe, i.e. ending of one Kalpa and the beginning of another, demon Hayagriva attempted to become eternal by "swallowing" the Vedas as they escaped from Brahma's mouth, but the Vedas were restored by Vishnu's avatara, Matsya. Vishnu also killed other asuras like Hiranyakasipu and Ravana who tried to become immortal by obtaining boons from devas.
An alternate Hayagriva story was that Hayagriva (horse-headed) could only be killed by another Hayagriva. The Asura Hayagriva terrorized the three worlds. Somehow, while asleep, Vishnu was decapitated, and a horse head replaced his "human-like" head, and he was able to defeat Asura Hayagriva.