Suppose that $G$ is a finite group, acting via homeomorphisms on $B^n$, the closed $n$-dimensional ball. Does $G$ have a fixed point?

A *fixed point* for $G$ is a point $p \in B^n$ where for all $g \in G$ we have $g\cdot p = p$.
Notice that the answer is "yes" if $G$ is cyclic, by the Brouwer fixed point theorem. Notice that the answer is "not necessarily" if $G$ is infinite. If it helps, in my application I have that the action is piecewise linear.

First I thought this was obvious, then I googled around, then I read about Smith theory, and now I'm posting here.