Suppose to have two Banach spaces $E$ and $F$, with $E$ reflexive. Suppose to have a continuous map $T:E \to F$ which maps bounded subsets into precompact subsets. $T$ is not assumed to be linear. Finally, suppose to have a sequence $(e_n)_n\subset E$ weakly convergent to $e\in E$.
Then, up to subsequences, $T(e_n)$ converges strongly to $T(e)$.
Is it correct? How can I prove it? Can I drop the reflexivity assumption?