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Let $K_1$, $K_2$ be compact Hausdorff spaces and $E_1, E_2$ be Banach spacesnot containing a copy of $c_0$. We establish lower estimates of theBanach-Mazur distance between the spaces of continuous functions$\mathcal{C}(K_1, E_1)$ and $\mathcal{C}(K_2, E_2)$ based on the ordinals$ht(K_1)$, $ht(K_2)$, which are new even for the case of spaces of real valuedfunctions on ordinal intervals. As a corollary we deduce that $\mathcal{C}(K_1,E_1)$ and $\mathcal{C}(K_2, E_2)$ are not isomorphic if $ht(K_1)$ issubstantially different from $ht(K_2)$.

Isomorphisms of $\mathcal{C}(K, E)$ spaces and height of $K$