Extensions of Nash equilibria to games with infinite strategies
Yes, there is such a setting. The result is that


  If each player's strategy space is
  
  
  convex
  compact
  
  
  and if payoffs are continuous then there exists at least one Nash equilibrium (possibly in mixed strategies).


This holds even when the set of possible actions is uncountably infinite. If one additionally assumes that payoffs are quasiconcave then the best-response correspondence will be convex even when we restrict attention to pure strategies so that we are then guaranteed to have at least one equilibrium in pure strategies in such a game.

I believe the original reference here is 


I.L. Glicksberg. "A further generalization of the Kakutani fixed
point theorem, with application to Nash equilibrium points."
Proceedings of the American Mathematical Society, 3(1):170–174, 1952.


The treatment in Glicksberg's paper, though, does not  seem very accessible. A good starting reference is more likely to be section 1.3 of Fudenberg &amp; Tirole's book "Game Theory".