  The aim of this article is to extend results of M.~Popov and second named
author about orthogonally additive narrow operators on vector lattices. The
main object of our investigations are an orthogonally additive narrow operators
between lattice-normed spaces. We prove that every $C$-compact
laterally-to-norm continuous orthogonally additive operator from a
Banach-Kantorovich space $V$ to a Banach lattice $Y$ is narrow. We also show
that every dominated Uryson operator from Banach-Kantorovich space over an
atomless Dedekind complete vector lattice $E$ to a sequence Banach lattice
$\ell_p(\Gamma)$ or $c_0(\Gamma)$ is narrow. Finally, we prove that if an
orthogonally additive dominated operator $T$ from lattice-normed space $(V,E)$
to Banach-Kantorovich space $(W,F)$ is order narrow then the order narrow is
its exact dominant $\ls T\rs$.
