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We show that the Gurarij space $\mathbb{G}$ has extremely amenableautomorphism group. This answers a question of Melleray and Tsankov. We alsocompute the universal minimal flow of the automorphism group of the Poulsensimplex $\mathbb{P}$ and we prove that it consists of the canonical action on$\mathbb{P}$ itself. This answers a question of Conley and T\"{o}rnquist. Weshow that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ isextremely amenable. Similarly, the pointwise stabilizer of any closed properbiface of the unit ball of the dual of the Gurarij space (the Lusky simplex) isextremely amenable.  These results are obtained via several Kechris-Pestov-Todorceviccorrespondences, by establishing the approximate Ramsey property for severalclasses of finite-dimensional Banach spaces and function systems and theirversions with distinguished contractions. This is the first direct applicationof the Kechris-Pestov-Todorcevic correspondence in the setting of metricstructures. The fundamental combinatorial principle that underpins the proofsis the Dual Ramsey Theorem of Graham and Rothschild.

The Ramsey property for Banach spaces and Choquet simplices