Baxter's Spline spacecraft are grown, and at least partially sapient; Mofitt's 'Genesis Quest' has interstellar, nay intergalactic trees, though they don't have much of a propulsion system. I remember a short story where the main characters were space-living organisms where the adults were ships, and the young passengers.
My principal difficulty is with an organic drive system that is at the same time reasonably efficient and fits into present-day physics. Then again, the drive might be dead matter (equivalent to the mineral shell of a coral polyp), though this removes the auto-reparing function that's a living vessel's most positive aspect.
Of course there are any number of ships which are cyborgs, with some part (generally the control system or parts of life support) living, while others are manufactured. Ships that sing, mutate, eat asteroids…
My concept is that the ships are principally grown from trees, which have the unique ability to take on useful properties from things which they come into contact with, such as metals and other materials. The propulsion system is a little bit special, so I won't reveal too much about that.