Is weakly continuous function with values in Banach space strongly measurable？

Question

Let $X$ be a non-separable Banach space and a map $f$ from real line into $X$ is called weakly continuous if it is continuous with respect to the weak topology of $X$. A function with values in $X$ is called strongly measurable if it is weakly measurable and separably valued(Pettis theorem). It seems that a weakly continuous function is automatically weakly measurable. Is it also separably valued(i.e., its range lies in a closed separable subspace of $X$)?