  Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbert
space with dim$(H_\mathbb{R})\geq 2$. We prove that the space $C(K,H)$ of all
$H$-valued continuous functions on $K$, equipped with the supremum norm,
satisfies the Mazur--Ulam property, that is, if $Y$ is any real Banach space,
every surjective isometry $\Delta$ from the unit sphere of $C(K,H)$ onto the
unit sphere of $Y$ admits a unique extension to a surjective real linear
isometry from $C(K,H)$ onto $Y$. Our strategy relies on the structure of
$C(K)$-module of $C(K,H)$ and several results in JB$^*$-triple theory. For this
purpose we determine the facial structure of the closed unit ball of a real
JB$^*$-triple and its dual space.
