Let $V$ be a connected smooth complex projective curve of negative Euler characteristic. Can there exist a connected smooth complex algebraic curve $U$ such that there is a non-constant holomorphic map $U\to V$ but no non-constant holomorphic map from the compactification of $U$ to $V$? Note that we are not merely asking that the map $U\to V$ does not extend.

EDIT: the question considers the smooth compactification of $U$.