On the group of infinite $p$-adic matrices with integer elements

Let $G$ be an infinite-dimensional real classical group containing the complete unitary group (or the complete orthogonal group) as a subgroup. Then $G$ generates a category of double cosets (train), and any unitary representation of $G$ can be canonically extended to the train. We prove a technical lemma on the complete group $\mathrm{GL}$ of infinite $p$-adic matrices with integer coefficients; this lemma implies that the phenomenon of an automatic extension of unitary representations to a train is valid for infinite-dimensional $p$-adic groups.