This is a follow-up to the question Connected vertex-transitive graph with the fixed-point property. In particular, it is based on a comment by user bof.

Let $G = (V,E)$ be a graph with $V$ infinite. Suppose $G$ is vertex-transitive, i.e., for every pair of vertices $u,v \in V$ there is an automorphism $\gamma$ of $G$ for which $\gamma(u) = v$. Is it possible that every automorphism of $G$ has at least one fixed vertex?

The previous question dealt with $V$ finite, for which the answer is no (as long as $\#V \geq 2$) by a simple group theory argument. But here it does not seem possible to use a group theory argument-- at least not so easily-- because there are transitive actions on infinite sets in which every permutation has a fixed point.