Of course, 99% of the commenters gave a loud YES, partly because 99% of them probably are already writing without getting paid. Writers write, and as much as we'd like to sell books and make some money, that's not why most of us do it. We do it because it's fun, it's challenging, and we can't really help ourselves.
A more interesting question, to my mind, is whether someone would be willing to stop writing if they were paid a million dollars (to pick a random number).
I don't think I would take the money. I'm sure there are ways to game the question such that I could still write something, but to stay within the question's intent, I'm pretty sure I wouldn't give up writing for a million bucks.
How about you? Is there anything you'd still do even if you didn't get paid for doing it... and would you stop for a million dollars?