Let $X$ be a locally compact Hausdorff space, such that $C_0(X)$ is an injective Banach space, *i.e.* a $\mathfrak{P}_\lambda$ space for some $\lambda\geq 1$.

Is it true that $X$ is compact?

If additionally $X$ is a locally compact group, is it true that $X$ is finite?

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