Arguably, the most well-known or famous bio-ship is Moya, loving home to refugees in the Uncharted Territories in Farscape (1999 - 2004). Moya is a gentle space-whale of sorts, one with emotions and high intelligence. She also feels pain when attacked. Moya, like all life-forms, is also able to reproduce. Early in Farscape's run, she produces a child, Talyn, who -- because of genetic meddling -- is a Peacekeeper battleship.
Moya also boasts a strong connection to "Pilot," a gentle, turtle-like alien who serves as the bio-ship's link to the humanoid world. Pilot often feels Moya's emotions, and is an advocate for her well-being with the biped crew.
In this 1990 story, Gomtuu is an acorn, or sea-shell-shaped bio-ship who is able to communicate on an empathic level with Tam Elbrun (Harry Groener), an unstable Betazoid. Gomtuu explains to Elbrun (and subsequently the NCC-1701-D crew...) that the bio-ship feels intense guilt over the accidental death of its humanoid crew. In addition, Gomtuu -- like Tam himself -- is lonely. The episode is about two disparate beings finding perfect companionship in one another.
In recent years, this status as a distinctive and separate life-form (rather than as a machine, for instance) has become more plain, especially in episodes such as "The Doctor's Wife," during the era of the Eleventh Doctor (Matt Smith).
In this tale, the TARDIS's soul or spirit is transferred into a humanoid woman, one who re-contextualizes the Doctor's adventures in a new and different way. She suggests she "stole" the Doctor from Gallifrey, not vice-versa, for instance. She also reveals a real affection and love for the Doctor (and a dislike or distrust of some of the "strays" he brings aboard the ship in the form of his companions.)