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To appear in Annales Mathématiques Blaise Pascal.International audienceWe prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $\O$ and any point $x \in \O$, any discrete group generated by a finite number of automorphisms of $\O$, which displace $x$ at a distance less than $\varepsilon_n$, is virtually nilpotent

Un lemme de Kazhdan-Margulis-Zassenhaus pour les géométries de Hilbert