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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)

Let $E \subseteq F$ be an (isometric) inclusion of Banach spaces, with $\dim F/E$ finite, and let $E^*_{=1}$, E^*_1$,$F^*_{=1}$F^*_1$ denote the closed unit spheres balls of their respective duals. Then the restriction map $F^* F^*_1 \to E^*$ E^*_1$admits a weak$^*$-continuous sectionfrom$E^*_{=1}$to$F^*_{=1}$.. This must be known - can anyone provide a reference? (am not a Banach space theorist) EDIT: changed "unit balladded "in the original question to finite dimension" unit sphere"hypothesis. 3 deleted 5 characters in body I need, and (unless I am seriously mistaken) can prove, the following: Let$E \subseteq F$be an (isometric) inclusion of Banach spaces, and let$E^*{=1}$, E^*_{=1}$, $F^*{=1}$ 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}$E^*_{=1}$ to $F^*{=1}$.F^*_{=1}\$.