The Runge approximation theorem for open Riemann Surfaces says the following. If $X$ is an open Riemann surface and $Y \subset X$ is a Runge subset, i.e., $X \setminus Y$ has no relatively compact connected components, then any holomorphic function on $f:Y \to \mathbb{C}$ can be approximated, uniformly on compact subsets of $Y$, by holomorphic functions on $X$.
My question is the following. Say $f$ on $Y$ is nonvanishing, i.e., $f:Y \to \mathbb{C}\setminus \{0\}$. Can $f$ be approximated, uniformly on compact subsets of $Y$, by nonvanishing holomorphic functions $X \to \mathbb{C}\setminus \{0\}$?
Obviously this problem is easy if I could lift the function $f$ with respect to the exponential map, but in general we can't do that. We can obviously locally lift, i.e. lift $f$ on neighbourhoods of every point of $Y$, but I'm not sure how that could be useful. Does anyone have any ideas?
Edit: to avoid topological obstructions, we may assume that $f$ can be extended to a continuous nonvanishing function on $X$.