Let $C_p$ b the cyclic group of order $p$, with $p$ a prime.  

Is is possible for $C_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points?

Standard Smith Theory implies that for this to be possible, $H_*(X;\mathbb Z/p)$ would have to be infinite dimensional.  (So, e.g., with $p=2$, $X$ might be homotopic to an infinite wedge of $\mathbb RP^2$s.)

I am rather hoping that an example like this exists.  But a proof that this can't happen would be fine too!