The ability to gate-induce superconductivity by electrostatic charge accumulation is a recent breakthrough in physics and nanoelectronics. With the exception of LaAlO3/SrTiO3 interfaces, experiments on gate-induced superconductors have been largely confined to resistance measurements, which provide very limited information about the superconducting state. Here, we explore gate-induced superconductivity in MoS2 by performing tunnelling spectroscopy to determine the energy-dependent density of states (DOS) for different levels of electron density n. In the superconducting state, the DOS is strongly suppressed at energy smaller than the gap Δ, which is maximum (Δ ~2 meV) for n of ~1 × 1014 cm-2 and decreases monotonously for larger n. A perpendicular magnetic field B generates states at E < Δ that fill the gap, but a 20% DOS suppression of superconducting origin unexpectedly persists much above the transport critical field. Conversely, an in-plane field up to 10 T leaves the DOS entirely unchanged. Our measurements exclude that the superconducting state in MoS2 is fully gapped and reveal the presence of a DOS that vanishes linearly with energy, the explanation of which requires going beyond a conventional, purely phonon-driven Bardeen-Cooper-Schrieffer mechanism.