The short answer is: Yes, but it's very difficult to learn. Research has shown that adults are capable of developing perfect pitch (more precisely called: absolute pitch).
However, the training is slow, tedious and unreliable. To reach a fully-developed sense of perfect pitch (meaning you can instantly identify all the notes present in a piece of music you hear) would likely take years. Musicians who start down this route typically develop a very weak and limited sense of perfect pitch after months of hard work, and give up there. Perfect pitch is a neat party trick, and it can be useful.
However if you are interested in learning perfect pitch for practical musical purposes such as playing by ear, improvising, or writing your own music, then relative pitch is a far easier and more rewarding skill to learn. There are plenty of resources for developing relative pitch (a skill which includes interval recognition, solfa, chord and chord progressions, and scales) and many adults can reach a very useful level of ability within a few months.
The only downside of relative pitch compared with perfect pitch is that you can't directly identify the key of a song, so you're still reliant on having some known point of reference (e.g. a single known note, or being told the key or starting chord) to apply your relative pitch on an instrument. This isn't a huge limitation.
In practice, in most musical situations, you do know (or can easily find out) at least one note or chord which, combined with your sense of relative pitch, reveals all the rest. If you are determined to be fully independent and recognise notes by name without any clue from the musical situation, then the approach I recommend is to learn a single "reference pitch".
For example, guitarists might choose E because they're used to tuning their two E guitar strings. Or concert players might choose A440. Learning a single reference note like this is far easier than developing a full sense of perfect pitch, and by combining that single reliable reference point with a well-developed sense of relative pitch you can accomplish everything that's possible with perfect pitch.