Behaviour of holomorphic automorphisms on equicontinuous subsets of the space ${\mathcal{C}} (\Omega ,E)$

Consider a compact Hausdorff topological space $\Omega$, a $\text{JB}^{\ast }$-triple $E$ and $F: = {\mathcal{C}}(\Omega , \, E)$, the $\text{JB}^{\ast }$-triple of all continuous $E$-valued functions $f\colon \Omega \to E$ with the pointwise operations and the norm of the supremum. Let ${\mathsf{G}}$ be the group of all holomorphic automorphisms of the unit ball $B_{F}$ of $F$ that map every equicontinuous subset lying strictly inside $B_{F}$ into another such a set. The real Banach-Lie group ${\mathsf{G}}$ and its Lie algebra are investigated. The identity connected component of ${\mathsf{G}}$ is identified when $E$ has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case $E={\mathbb{C}}$.