Let $K$ be a compact, Hausdorff space and $X$ be a Banach space. By $C(K,X)$ we denote the Banach space of all continuous functions $f : K \to X$, equipped with the supremum norm: \begin{align} \|f\|_\infty = \sup_{k \in K} \|f(k)\|. \end{align}
Suppose $K$ is scattered. Is $(C(K,X))^*$, the dual space of $C(K,X)$, isomorphic to $\ell_1(I, X^*)$, for some non-empty set $I$?
I know this is true if $X = \mathbb{R}$, but despite my best efforts was unable to find a proof of this more general case.
A reference to a proof in the affirmative case, or a counter-example in the negative would be most appreciated.