Almost everywhere convergence of subsequence in Bochner space

Question

Does the familiar theorem that convergence in $L^p$ implies almost everywhere convergence along a subsequence also hold in Bochner spaces?

I.e., for a Banach space $B$ does $f_n\to f$ in $L^p(\Omega;B)$ imply the existence of a subsequence such that $f_{n_k}(\omega)\to f(\omega)$ in $B$ for almost every $\omega$?