The dynamics of a mobile quantum impurity in a degenerate Fermi system is a fundamental problem in many-body physics. The interest in this field has been renewed due to recent ground-breaking experiments with ultracold Fermi gases1,2,3,4,5. Optical creation of an exciton or a polariton in a two-dimensional electron system embedded in a microcavity constitutes a new frontier for this field due to an interplay between cavity coupling favouring ultralow-mass polariton formation6 and exciton–electron interactions leading to polaron or trion formation7,8. Here, we present cavity spectroscopy of gate-tunable monolayer MoSe2 (ref. 9) exhibiting strongly bound trion and polaron resonances, as well as non-perturbative coupling to a single microcavity mode10,11.