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".