The standard proofs of the existence of Nash equilibria in game theory all use either Brouwer's or Kakutani's fixed point theorem. See for example Nash's 1951 paper "Non-Cooperative Games," where he defines his equilibrium notion and gives the Brouwer-based proof.
Recent complexity-theory results by Daskalakis, Papadimitriou, etc. showing PPAD-completeness of computing Nash equilibria mean that in some sense a fixed point theorem (or equivalent) is necessary to prove existence of Nash equilibria.
A fixed point theorem like the Banach one does not in general apply to this problem, because there can be multiple equilibria.