  Dark matter could accumulate around neutron stars in sufficient amounts to
affect their global properties. In this work, we study the effect of a specific
model for dark matter -- a massive and self-interacting vector (spin-1) field
-- on neutron stars. We describe the combined systems of neutron stars and
vector dark matter using Einstein-Proca theory coupled to a nuclear-matter
term, and find scaling relations between the field and metric components in the
equations of motion. We construct equilibrium solutions of the combined
systems, compute their masses and radii and also analyse their stability and
higher modes. The combined systems admit dark matter (DM) core and cloud
solutions. Core solutions compactify the neutron star component and tend to
decrease the total mass of the combined system. Cloud solutions have the
inverse effect. Electromagnetic observations of certain cloud-like
configurations would appear to violate the Buchdahl limit. This could make
Buchdahl-limit violating objects smoking gun signals for dark matter in neutron
stars. The self-interaction strength is found to significantly affect both mass
and radius. We also compare fermion Proca stars to objects where the dark
matter is modelled using a complex scalar field. We find that fermion Proca
stars tend to be more massive and geometrically larger than their scalar field
counterparts for equal boson masses and self-interaction strengths. Both
systems can produce degenerate masses and radii for different amounts of DM and
DM particle masses.
