A topological group $G$ is extremely amenable if every continuous action of $G$ on a compact Hausdorff space has a fixed point. The Kechris-Pestov-Todorčević (KPT) theorem identifies extreme amenability as a Ramsey-theoretic phenomenon: a closed subgroup $G$ of $\mathrm{Sym}(\mathbb{N})$ is extremely amenable if and only if $G = \mathrm{Aut}(M)$, where $M$ is the Fraïssé limit of a Fraïssé class of relational structures with the Ramsey Property. By analogy with the topological group case, let’s call a category $C$ extremely amenable if every presheaf on $C$ valued in compact Hausdorff spaces has a global element. Under some mild hypotheses on $C$, $C$ is extremely amenable if and only if it satisfies a natural categorical variant of the Ramsey Property. This point of view provides a unified explanation for the appearance of the Ramsey property in the KPT theorem and in the existence of generalized indiscernibles in model theory.