I need, and (unless I am seriously mistaken) can prove, the following:

Let $E \subseteq F$ be an (isometric) inclusion of Banach spaces, and let $E^*_{=1}$, $F^*_{=1}$ denote the unit spheres of their respective duals. Then the restriction map $F^* \to E^*$ admits a weak$^*$-continuous section from $E^*_{=1}$ to $F^*_{=1}$.

This must be known - can anyone provide a reference? (am not a Banach space theorist)

EDIT: changed "unit ball" in the original question to "unit sphere".