Polyadic rings of $p$-adic integers

In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $(m,n)$-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $(m,n)$-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.