A well-known result of Andreatta and Wisniewski says: Let $X$ be a projective complex manifold whose tangent bundle $T_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to projective space $\mathbb{P}^n$ for some $n$. Furthermore, the bundle $\mathscr{E}$ is either $\mathscr{O}(1)^{\oplus r}$ or $T_{\mathbb{P}^n}$ itself.

I would like to know if a slightly stronger statement is known/true.

Suppose there is an ample vector bundle $\mathscr{E}$ and a non-zero

map$$f: \mathscr{E}\to T_X.$$ Is $X$ necessarily isomorphic to $\mathbb{P}^n$ for some $n$?

Of course if $\text{im}(f)$ is a vector bundle, the result follows by Andreatta-Wisniewski, as quotients of ample vector bundles are ample.

**Remarks (Added later).** The statement is true if $\mathscr{E}$ is a line bundle, or if $X$ is a curve or surface. I can also show that the existence of a non-zero map $\mathscr{E}\to T_X$ with $\mathscr{E}$ ample implies $X$ is birational to an (etale) $\mathbb{P}^k$-bundle over a lower-dimensional variety (and in particular, is uniruled). The answer of pgraf gives a reference showing the statement is true if $\text{Pic}(X)$ has rank $1$.

The paper "Galois coverings and endomorphisms of projective varieties" by Aprodu, Kebekus, and Peternell, which pgraf has referenced and which covers the case of Picard rank $1$ only uses the hypothesis in Corollaries 4.9 and 4.11, as far as I can tell. In particular, many of the other arguments in Section 4 of that paper hold true; for example, one knows that if $T$ is a family of rational curves in $X$ of minimal degree, any split curves must land in the singular locus of $\text{im}(f)$. Analyzing the arguments of that paper seem to give several other cases, e.g. if $f$ has generic rank equal to $\text{dim}(X)$.