  We deal with isomorphic Banach-Stone type theorems for closed subspaces of
vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or
$\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over
$\mathbb{F}$ with a certain parameter $\lambda(E_i)>1$, which in the real case
coincides with the Schaffer constant of $E_i$, let $K_i$ be a locally compact
(Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of
$\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary
$\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We
show that if there exists an isomorphism $T:\mathcal{H}_1 \rightarrow
\mathcal{H}_2$ with $\Vert T \Vert \cdot \Vert T^{-1} \Vert<\min \lbrace
\lambda(E_1), \lambda(E_2) \rbrace$, then $\mathcal{Ch}_{\mathcal{H}_1} K_1$ is
homeomorphic to $\mathcal{Ch}_{\mathcal{H}_2} K_2$. Next we provide an
analogous version of the weak vector-valued Banach-Stone theorem for subspaces,
where the target spaces do not contain an isomorphic copy of $c_0$.
