The authors study so-called weighted set-valued maps (in their terminology called \(m\)-multivalued maps), i.e. compact-valued upper semicontinuous mappings \(F: X\to Y\) between compact spaces whose values are finite disjoint unions of finitely many components, acyclic with respect to the Čech homology with coefficients in a field \(\mathbb{F}\) and endowed with a weight, i.e., a \(\mathbb{F}\)-valued multiplicity function \(m\) defined on the graph of \(F\) and satisfying some regularity properties. Extending the Čech homology functor for such maps, the authors introduce the concept of Lefschetz number and investigate the transfer homomorphism. Finally, they introduce the notion of an \(\mathbb{F}\)-simplicial space (showing that, in particular, compact absolute neighborhood retracts are \(\mathbb{F}\)-simplicial) and prove a variant of the Lefschetz fixed point theorem for \(m\)-multivalued maps. The involved proof relies on the notion of a generalized weak approximative system which helps to proceed on the level of chains of homology groups. The stated and proved results generalize by far the existing Lefschetz type theorems for set-valued compact maps.