I am interested in the following question:

Given a geometrically connected smooth rigid analytic space $X$ over a non-archimedean field $k$, is it always possible to find an affinoid open covering, where each element of the covering is geometrically connected?

Or perhaps more generally: is it true that any connected affinoid open subset of $X$ is geometrically connected? I don't imagine that this is true, but I can't think of a counterexample.

For example, if $X$ were a geometrically connected smooth scheme over a field, then because $X$ is normal, $X$ is actually geometrically irreducible. Therefore any connected open subset is automatically geometrically irreducible, so geometrically connected. In particular, the question would be true in this case.

Any ideas, counterexamples, or insight would be very much appreciated.