Wild ramification kinks

Given a branched cover f : Y → X between smooth projective curves over a non-archimedean mixed-characteristic local field and an open rigid disk D ⊂ 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.