  We construct $p$-adic analogs of operator colligations and their
characteristic functions. Consider a $p$-adic group $G=GL(\alpha+k\infty,
Q_p)$, its subgroup $L=O(k\infty,Z_p)$, and the subgroup $K=O(\infty,Z_p)$
embedded to $L$ diagonally. We show that double cosets $\Gamma= K\setminus G/K$
admit a structure of a semigroup, $\Gamma$ acts naturally in $K$-fixed vectors
of unitary representations of $G$. For each double coset we assign a
'characteristic function', which sends a certain Bruhat--Tits building to
another building (buildings are finite-dimensional); image of the distinguished
boundary is contained in the distinguished boundary. The latter building admits
a structure of (Nazarov) semigroup, the product in $\Gamma$ corresponds to a
point-wise product of characteristic functions.
