A metric space $(X,d)$ is isometrically homogeneous if its isometry group acts transitively on points, i.e., for every $x,y \in X$ there is an isometry $\varphi:X\to X$ with $\varphi(x) = y$. I'd like to know an example of a compact isometrically homogeneous metric space which is not a manifold (a space with finitely many points counts as a 0-dimensional manifold).
Googling a bit I've discovered enough recent literature on this general subject to be sure there must be classical examples known to experts, but I haven't managed to find them written down. For example, Theorem 1.2 of this paper implies:
A compact isometrically homogeneous metric space is a finite-dimensional manifold if and only if it is locally contractible.
So equivalently, I'd like an example of a compact isometrically homogeneous metric space which is not locally contractible.
Added: Pete and Neil both gave very nice answers. I'm accepting Neil's since, as Pete points out, it essentially contains Pete's answer as a special case.