  We present new constraints on three different models, the so-called
universal, $B-L$ and $L_\mu-L_\tau$ models, involving a yet to be observed
light vector $Z'$ mediator, by exploiting the recent observation of coherent
elastic neutrino-nucleus scattering (CE$\nu$NS) in argon and cesium-iodide
performed by the COHERENT Collaboration. We compare the results obtained from a
combination of the above data sets with the limits derived from searches in
fixed target, accelerator, solar neutrino and reactor CE$\nu$NS experiments,
and with the parameter region that could explain the anomalous magnetic moment
of the muon. We show that for the universal and the $B-L$ models, the COHERENT
data allow us to put stringent limits in the light vector mediator mass,
$M_{Z'}$, and coupling, $g_{Z'}$, parameter space.
