You are allowed to keep thinking as long as you have time on your clock. Isn't really considered good sportsmanship but is legal.
Recent instances I saw was an adult was in an almost-lost position with over an hour on the clock while his opponent had 10 minutes. He let his clock run down to nothing and then played quickly before finally let the clock run to zero in a lost (mate in 2) position. He got mocked for this in the local forums.
Also common for a kid to do a blunder and then sit there sad/crying for an hour. You try to encourage them to resign though.
Source: Am Chess Player/Organizer/arbiter.
I am not a chess player but I have never heard of referees stopping the game if you don't "give up" in a losing position and have extra time. That sounds ridiculous but I would love to know if it applies in certain tournaments and the reasoning behind it.