  Given a branched cover $f:Y\to X$ between smooth projective curves over a
non-archimedian mixed-characteristic local field and an open rigid disk
$D\subset X$, we study the question under which conditions the inverse image
$f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in
an analytic family, is this true at least for some member of the family? Our
main result gives a criterion for this to happen.
