Lones Smith offers the following proof: if you have countably many players, arbitrarily enumerate them and call the original game with this enumeration $G$. Let $G_n$ be a game where the actions of players $k>n$ are fixed arbitrarily. By Glicksburg, $G_n$ has an equilibrium -- call it $x(n)$. Let $x$ be a limit point of the sequence $(x(n))_{n=1}^\infty$. Fix any player $i$. Now $x_i(n)$ is a best response to $x_{-i}(n)$ in $G_n$ for each $n$; to pass to limits, we appeal to the Theorem of the Maximum and continuity of the utility functions [1]. That shows the action $x_i$ is a best response to $x_{-i}$ in $G$. So $x$ is an equilibrium of $G$.

[1] This assumes a topology on the action space at least as coarse as the product topology, so that whenever $x(n)$ converges to $x$ in the product topology, it converges in the topology in question. The utilities should be continuous under this topology for actions. This is a big assumption and easily violated!