QUESTION RETRACTED - My original argument was fundamentally mistaken (mixing up lower and upper semi-continuity). Sorry (and thanks for the useful comments)
I need, and believe (unless I am seriously mistaken) 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 weak$^\*$-continuous section from $E^*_1$ to $F^*_1$.
On the unit sphere this section gives you norm-preserving extensions.
This must be known - can anyone provide a reference? (am not a Banach space theorist)
EDIT: added "finite dimension" hypothesis.