Let $S$ be a complex manifold, which for our purposes we can take to be a small ball in $\mathbb C^n$. Let $p : E \to S$ be a holomorphic vector bundle over $S$, and let $\pi : X \to S$ be a family of compact complex manifolds over $S$ (i.e. $X$ is a complex manifold, and $\pi$ is a holomorphic proper submersion).

We can pull the vector bundle $E$ back to $X$. This gives a vector bundle $\pi^*E \to X$, which we can regard as a fiber bundle $\pi^*E \to S$ over $S$.

Does the relative tangent bundle of the fiber bundle $\pi^*E \to S$ split holomorphically as the sum of the relative tangent bundles of $X \to S$ and $E \to S$? In symbols, do we have $T_{\pi^*E/S} = T_{X/S} \oplus T_{E/S}$ (the latter two bundles having been pulled back to $\pi^*E$)?

For a situation where such a thing appears naturally, take $E \to S$ to be the Hodge bundle of the fibers of $X$, so it has fibers $E_s = H^k(X_s,\mathbb C)$. Then $\pi^*E \to S$ has fibers $X_s \times H^k(X_s,\mathbb C)$.