Dedicated to the memory of Adriaan Cornelis Zaanen, the one of the great founder of the theory of vector lattices Abstract. We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow.Mathematics Subject Classification (2010). Primary 46B99; Secondary 46G12.