A member of the Dominion BB, NiceGuyEddie, complied this list, with the help of other BB members.
Living Ships We Have Known
Television:
StarTrek TNG - Pilot Episode "Farpoint" - a living ship that creates things upon request.
StarTrek TNG - Episode " Tin Man" - a living symbiotic ship that has lost its crew.
Babylon5 - Vorlon ships (biological) and Membari ships (biomechanoid)
Kitt - sentient car
Lexx - a living ship
Cartoons:
Speed Buggy - a talking dune buggy
Books:
Robin Hobb fantasy series: "Ship Of Magic" - Mad Ship, a living sea-faring vessel.
Series about ships with human consciences placed in them:
Anne McCaffrey - The Ship Who Sang
Anne McCaffrey and Margaret Bell - PartnerShip
Anne McCaffrey and Mercedes Lackey - The Ship Who Searched
S.M. Stirling - The Ship Avenged
John Varley - Titan, Wizard, Demon
Arthur C. Clarke - Rama, a ship with biological machines that act like DRDs.
Harlan Ellison - The Human Operators (also turned into an episode of The New Outer Limits) - a sentient spacebattle cruiser