Let $X$ and $Y$ be complex manifolds and let $\pi \colon X \to Y$ be holomorphic proper surjective submersion. Therefore, $\pi$ is a fiber bundle whose fibers are complex manifolds.
My question: Does there exists for every $x \in X$ a neighborhood $U \subset X$ of $x$ with a vertical holomorphic vector field $v$ on $U$, i.e. a vector field $v$ on $U$ such that $\pi_{\ast} v = 0$?
Equivalently, one could ask for local coordinate $z \colon U \to \mathbb{C}$ such that $\partial_z$ is vertical. I know that the result holds in the smooth category. However, I am not sure whether one can choose the desired vector field to be holomorphic.