Let $S\subset\mathbb{R}^n$ be compact, $\alpha,\beta\in(0,1)$, $\alpha>\beta$ and $X$ a Banach space. Under which assumptions on $X$ is the embedding $$C^\alpha(S;X)\subset C^\beta(S;X)$$ compact?

The For $X=\mathbb{R}^N$ compactness holds and is a consequence of Ascoli-Arzela's theorem. The above question seems to boil down to whether there's a Banach space valued version of Ascoli-Arzela's theorem. References are welcome. Thanks.