Can we hope for application of Etale cohomology techniques in proving results concerning semialgebraic subsets of $\mathbb{Q}_p^n$?

Recall that semialgebraic subsets are obtained from $p$-adic algebraic varieties using boolean connectives and coordinate projection.

The results I am interested in are a sort of algebraic topology over the $p$-adics. But of course one cannot use the standard topology on the $p$-adics since it is totally disconnected. So I hope to show for instance that a ball with one smaller ball removed cannot be diffeomorphic to a ball, etc .