Let $X$ be a topological space. $X$ is said to be homogeneous if for every $x$ and $y\in X$ there is an self-homeomorphism $f$ of $X$ such that $f(x)=y$. Further, $X$ is said to have the fixed-point property if every endomorphism of $X$ has a fixed-point. My question is:
Is there a example of a topological space which has the fixed-point property but is also homogeneous (that isn't the one-point space)?
The only examples of fixed-point spaces I know are compact, convex subspaces of Euclidean space which are guaranteed by the Brouwer Fixed-Point Theorem. These spaces are not homogeneous since we cannot drag a boundary point into the interior by a homeomorphism. The examples I know of homogeneous spaces are topological groups and unit spheres neither of which have the fixed-point property.
I attempted to show the properties are incompatible, but there doesn't seem to be an inherent contradiction in the two properties. I do know that topological groups do not have the fixed-point property, so those are out of the search. But other than that, my search into an example or a proof of incompatibility has been fruitless.
Any help either way is appreciated.