I achieved a degree of absolute pitch (AP) as an adult, but of course this was not done in a controlled manner, so the scientific reliability of my case is implicitly questionable.
Anyway, here's what happened:
As a student who specialized in music in high school, and then pursued a Bachelor of Music degree in college, my path follows a pretty standard one in terms of trained classical musicians (who are not identified as "prodigies" VERY early in life).
Being around people who did posses AP meant that I was able to self-define as not possessing it, since I was incapable of doing the same kinds of mental-aural recall and identification that they were. My general "aural skills" were still very good, as any of my teachers will tell you; my aural skills professor asked me "Do you have perfect pitch?" in one of the first few weeks of class. It is common for a well-trained sense of relative pitch (RP) to be mistaken for AP when you meet a younger student whose skills seem to be beyond the level of RP training he/she is expected to have undergone. At high levels of musical training and skill, a musician that possesses AP is not capable of much more than a similarly-skillful musician that only possesses RP. Music theory is based almost entirely upon RP phenomena, so all musicians must learn it.
Anyway, about 1-2 years into my undergraduate degree, I started getting really interested in aural skills in general as well as the AP phenomenon, and started doing some hardcore middle-of-the-night "internet research" on the topic. There appeared to be some anecdotal evidence that AP could be learned, but at the same time most of the "training methods" being advertised looked like snake oil salesmen.
In the end, it turned out that there was a significant mental block I had to overcome simply to the fact that B and C (or any other two notes) sound different to one another. After that, there's a lot of almost meditative work to do to listen for those differences, while connecting back to repertoire that you know and can accurately recall; and finally comes the unending process of designing and studying aural skills exercises that specifically target your learning needs with increasing difficulty. That's the basic 3-step process in my experience.
Right now, my AP skill is at the point where I can produce or recognize any given pitch out of context with a very high degree of accuracy, but with non-instantaneous response times. When I am incorrect, I'm never off by more than a semitone. In these ways, I draw a lot of parallels between training AP and training RP -- in the sense that a student makes the same kinds of mistakes at lower levels of facility. Metaphorically speaking, it's as if the aural definition I can perceive of pitch is very sharp for some notes and sometimes blurry for others, but given enough time I can "figure it out" in my head for any note. Timbral differences throw in another variable that doesn't apply as much to RP training, at least for me. I have found all of this to improve with continued study.
For these reasons, I feel that "perfect pitch" (PP) is a useful term to use as distinct from AP--though colloquially they are seen as synonyms, I like to use AP as we do in this post, but use PP to refer to the "hyper-skillful" sense of AP that we see in people who picked it up intuitively at a very young age (honed by copious amounts of formal musical training).
In conclusion, I believe that the similarities I have experienced between AP and RP in my own musical (self-)education indicate that it is possible for anyone to achieve AP or even PP through training. There is of course an understanding that people are predisposed to musical skill based on their exposure to music during their upbringing, but this impacts RP awareness and training just the same as it does AP--and no one has ever said, "Oh, you'll never be able to play like Coltrane no matter how hard you work at it, because you're not genetically predisposed." There appear to be some societal preconceived notions that you can achieve any level of skill through dedicated training and hard work, but for some reason PP is an exception that "you must be born with". I see no reason for this to be anything but that; a preconceived notion. This is the closest one I can find to it at the moment, but I am aware of a study that shows that people without musical training, when singing their favorite pop songs a cappella, more often than not, are able to do so in the correct key. This indicates that most people have some intuitive pitch memory even if they don't realize it, which is one of the primary building blocks of AP training.
I haven't been engaged in training my own AP for a couple of years since I've been out of school, but I have some long-term goals of developing software and training methods for this, coupled of course with further honing of my own skills in a more controlled context. Consider this post a prelude to more scientific findings I hope to come up with in the future!