I am a PhD student in Mathematics. My advisor proposed me a problem 6 months ago, and I did a lot of progress on it. I have reached a point where I have quite substantial results (admittedly, adapting techniques from other areas, so not a great deal of originality). They can surely form a paper, though not a great one. The original question remains unanswered in its full generality, but I felt that a complete answer would be impossible from the very beginning, and now I have very concrete reasons showing that a complete answer is indeed very hard (in the sense that a general answer/formula might simply not exist). My advisor wants me to keep working on this, but I honestly don't know what to try anymore, and he did not suggest anything. I guess this is a common occurrence during a PhD, what should I do?
This sounds very familiar. Concrete advice in this situation: Focus on writing the paper.
You have some contribution, new arguments as to why the general problem is hard, progress on parts of the general problem. This really sounds like a possible paper - so write it and, if discussed with your advisor, submit it to the right journal.
Postpone your decision on further steps to this point. The situation may have changed completely until then - and if not, you will still be much better prepared to discuss the matter and to decide.
(Caveat: It may also happen, that, during writing, you'll find new ways to attack a more general version or find flaws/gaps in your current reasoning… anyway, writing the paper also help with this.)