Resolution of non-singularities for Mumford curves

Given a Mumford curve $X$ over $\bar{\mathbb Q_p}$, we show that for every semistable model $\mathcal X$ of $X$ and every closed point $x$ of this semistable model, there exists a finite \'etale cover $Y$ of $X$ such that every semistable model of $Y$ 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 $\bar{\mathbb Q_p}$ with isomorphic tempered fundamental groups are isomorphic over $\mathbb Q_p$.