Homotopy equivalence between a holomorphic vector bundle and the base space

Question

Given a smooth manifold $X$, there is a smooth homotopy equivalence between any smooth vector bundle $\pi: E\rightarrow X$ and $X$. It is usually proven by showing that the projection $\pi$ is a so-called smooth deformation retraction of $E$ on $X$. Is there a homotopy equivalence between a holomorphic vector bundle $p:F\rightarrow X$ on a complex manifold $X$ and the base space $X$? Is the canonical holomorphic projection $p$ of $F$ onto the complex manifold $X$ a holomorphic deformation retraction? Thanks for the advice.