Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or group cohomology.)

Assume there are only finitely many fixed points.

Can there be more than one?

Remarks: (1) If X is a manifold, this seems to hold, by the Lefschetz-Hopf fixed point theorem. But to generalize the proof to the general case, one would need to define the "index" of an isolated fixed point and I have no idea how to accomplish that, if one does not work with manifolds only. (2) The statement is true for graphs. (3) The statement is easily seen to be wrong for general contractible spaces, e.g. take a free action on $S^{\infty}$.