This is an extremely important question in vocal music -- and to a lesser for instrumentalists playing instruments, such as the violin family (or dobro!) which have infinitely variable pitch.
One of the major aspects of my vocal training was to develop my ability to retain pitch memory, and it was one of the things I really looked for in singers for one choir I once led, which performed a great deal of music either all "a cappela" (in the manner of the chapel; that is, without accompaniment) or with long stretches of it. It is, in fact, essential that singers of such works have a superb sense of relative pitch, and an equally superb sense of pitch memory. If you don't, my friend, when an accompaniment comes back in you have something between a catastrophe and a disaster.
Is this "perfect pitch"? There you run into a question of meaning. If by that you mean someone who can, on demand, sing a note which you designate as a concert A (or whatever), then if the ability is well enough developed (such as that soprano I mentioned in an earlier post) then it certainly is. On the other hand, if what you mean by "perfect pitch" you mean someone who is naive but who, on hearing a tone, can identify it as, say concert A, it isn't.
I very much doubt, since our notion of named pitch is clearly an agreed convention, that there could be such a thing as "perfect pitch" in the second sense above.
In the first sense, though, it is a trainable ability -- but like most other trainable abilities, one which some people have a better ability at to begin with than others.
I would add that the ability to discriminate between two tones very close together in frequency is also something which appears to be trainable, but which some people have better inherent ability in than others -- and it is a completely different ability from the ability to pick up relative pitch, and to remember pitch over time.