EXP-complete example
Every EXP-complete problem is not in P, by the time hierarchy theorem.  Many games are EXP-complete, you can look up some starting from here.

As a concrete example (albeit not a very simple one), you can consider the problem of emptiness of the intersection of deterministic tree automata. Formally, the input consists of $n$ DFTs $A_1,...,A_n$ (deterministic automata over trees), and the problem is to decide whether $L(A_1)\cap...\cap L(A_n)\neq \emptyset$.