Hi Flora,

it seems that many people have answered your question assuming that you're asking about becoming a research mathematician. But you can do a PhD in math and then go on to a different career, where you use your skills but don't necessarily go on to do original research. Many companies are hiring math PhDs and it seems to me that there are opportunities in very diverse fields. I'm not in the best position to discuss them ($p$-adic analysis doesn't seem to be very much in demand in the real world these days), but you should be able to find such information easily, maybe starting with the AMS?

And don't let yourself be discouraged by the behavior of your colleagues. It's true that some mathematicians spend all their waking hours thinking and talking about maths, and a few of them are very successful but these are really a minority. It's my opinion that most of those who display this behavior would really be better off if they had a more balanced lifestyle. There may be cultural factors here: in the protestant anglo-saxon world, it's important to be seen working all the time, and bad form for people to enjoy themselves.

Just do your own thing at your own pace. It's normal that you spend hours on a proof. Maths is not about being the first, but about having a deep understanding of what is going on. It certainly helps if you have flashes of insight, but there's nothing wrong about really understanding the situation first and finding the answer afterwards.

Concerning the funding, it's true that being a PhD student does not pay much and that can be a real problem. I don't know about the country where you are but in France, some companies have programs where you work for them part-time and do a PhD in the rest of your time, the PhD being in some applied domain that is then useful for the company.