Covering space of compact surface with free fundamental group

Question

Let $S$ be a compact connected surface. Does $S$ have a covering space $\tilde{S}$ such that the fundamental group of $\tilde{S}$ is the free group on $n$ generators with $n>1$ ?

I know that if we assume that the covering space is finite then there is no such covering, but what about if the covering space has an infinite fiber?