  We study two-dimensional materials where electrons are coupled to the vacuum
electromagnetic field of a cavity. We show that, at the onset of the
superradiant phase transition towards a macroscopic photon occupation of the
cavity, the critical electromagnetic fluctuations, consisting of photons
strongly overdamped by their interaction with electrons, can in turn lead to
the absence of electronic quasiparticles. Since transverse photons couple to
the electronic current, the appearance of non-Fermi-Liquid behavior strongly
depends on the lattice. In particular, we find that in a square lattice the
phase space for electron-photon scattering is reduced in such a way to preserve
the quasiparticles, while in a honeycomb lattice the latter are removed due to
a non-analytical frequency dependence of the damping $\propto |\omega|^{2/3}$.
Standard cavity probes could allow to measure the characteristic frequency
spectrum of the overdamped critical electromagnetic modes responsible for the
non-Fermi-liquid behavior.
