Suppose we have a finite map of compact Riemann surfaces $p : X\rightarrow\mathbb{P}^1_\mathbb{C}$. Suppose it is unramified above an open disk $D\subset\mathbb{P}^1_\mathbb{C}$. Thus, $p^{-1}(D)$ is a disjoint union of copies of $D$, each copy mapping homeomorphically onto $D$. Let $U$ be one of those copies. Thus, $p|_U : U\rightarrow D$ is a homeomorphism. Let $\overline{U}$ be the closure of $U$ inside $X$, and $\overline{D}$ the closure inside $\mathbb{P}^1_\mathbb{C}$. Must $p|_{\overline{U}}$ be a homeomorphism onto $\overline{D}$?
(Ie, is it possible for two boundary points of $U$ in $X$ to become identified in $\mathbb{P}^1_\mathbb{C}$?)
EDIT: By "$D$ is an open disk", I literally mean an open disk of some positive radius contained in the usual image of $\mathbb{C}$ in $\mathbb{P}^1_\mathbb{C}$. We may of course assume the disk is centered at 0.