  The CONNIE experiment is located at a distance of 30 m from the core of a
commercial nuclear reactor, and has collected a 3.7 kg-day exposure using a CCD
detector array sensitive to an $\sim$1 keV threshold for the study of coherent
neutrino-nucleus elastic scattering. Here we demonstrate the potential of this
low-energy neutrino experiment as a probe for physics Beyond the Standard
Model, by using the recently published results to constrain two simplified
extensions of the Standard Model with light mediators. We compare the new
limits with those obtained for the same models using neutrinos from the
Spallation Neutron Source. Our new constraints represent the best limits for
these simplified models among the experiments searching for CE$\nu$NS for a
light vector mediator with mass $M_{Z^{\prime}}<$ 10 MeV, and for a light
scalar mediator with mass $M_{\phi}<$ 30 MeV. These results constitute the
first use of the CONNIE data as a probe for physics Beyond the Standard Model.
