  We study self-gravitating bound states of a complex vector field, known as
Proca stars, with a new type of quartic-order self-interaction which does not
exist in the case of either a complex scalar field or a real vector field.
Depending on the sign of the coupling constant, this quartic self-interaction
can yield a distinct feature of Proca stars from the previously investigated
self-interaction of the vector field. We find that self-gravitating solutions
can be so compact that the photon sphere could form. However, we also show that
the self-interaction gives rise to a ghost instability for the stars whose
compactness is close that for the formation of a photon sphere, which might
invalidate the formation of the photon sphere.
