Is there a nontrivial commutative Hausdorff topological group that is extremely amenable?
Recall that a topological group is called extremely amenable if any continuous action on a compact Hausdorff topological space has a fixed point. For instance, it is known that no nontrivial locally compact group is extremely amenable, but some Polish groups, such as the group of order-preserving self-homeomorphisms of $[0,1]$, are extremely amenable.