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Let $K$ be a compact Hausdorff space and let $H$ be a real or complex Hilbertspace 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 theunit sphere of $Y$ admits a unique extension to a surjective real linearisometry 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 thispurpose we determine the facial structure of the closed unit ball of a realJB$^*$-triple and its dual space.

On the Mazur--Ulam property for the space of Hilbert-space-valued continuous functions