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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.

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 (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 closed unit balls of their respective duals. Then the restriction map $F^*_1 \to E^*_1$ admits a 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)

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".

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.

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".

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".