  We perform electric double-layer gating experiments on thin films of niobium
nitride. Thanks to a cross-linked polymer electrolyte system of improved
efficiency, we induce surface charge densities as high as $\approx 2.8 \times
10^{15}\,\mathrm{cm}^{-2}$ in the active channel of the devices. We report a
reversible modulation of the superconducting transition temperature (either
positive or negative depending on the sign of the gate voltage) whose magnitude
and sign are incompatible with the confinement of the perturbed superconducting
state to a thin surface layer, as would be expected within a na\"ive screening
model.
