Given a Mumford curve X over Q p , we show that for every semistable model X of X and every closed point x of this semistable model, there exists a finiteétale cover Y of X such that every semistable model of Y over X has a vertical component above x. We then give applications of this to the tempered fundamental group. In particular, we prove that two punctured Tate curves Q p with isomorphic tempered fundamental groups are isomorphic over Qp.