Let $M$ be a complex manifold and $X \subset M$ a complex submanifold. We may assume that $X$ is compact, if that's helpful.

Can we always find a neighbourhood $U$ of $X$ in $M$ together with a holomorphic map $r : U \to X$ which restricts to the identity map on $X$?

In the $C^\infty$-case, any tubular neighborhood gives such retractions. Of course, the tubular neighbourhood theorem may fail in the holomorphic case, but the existence of a retraction is *a priori* weaker.