Let $X$ be a smooth rigid space with an action of a finite group $G$ satisfying that $X/G$ is represented by a rigid space. We construct sheaves of $p$-adic Cherednik algebras on the small \'etale site of the quotient $X/G$, and study some of their properties. The sheaves of $p$-adic Cherednik algebras are sheaves of Fr\'echet-Stein $K$-algebras on $X/G$, which can be regarded as $p$-adic analytic versions of the sheaves of Cherednik algebras associated to the action of a finite group on a smooth algebraic variety defined by P. Etingof.
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