I need, and believe I can prove, the following:

Let $E \subseteq F$ be an (isometric) inclusion of Banach spaces, with $\dim F/E$ finite, and let $E^*_1$, $F^*_1$ denote the closed unit balls of their respective duals. Then the restriction map $F^*_1 \to E^*_1$ admits a weak$^*$-continuous section.

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

EDIT: added "finite dimension" hypothesis.