Studies of T Tauri discs inform planet formation theory; observations of variability due to occultation by circumstellar dust are a useful probe of unresolved, planet-forming inner discs, especially around faint M dwarf stars. We report observations of 2M0632, an M dwarf member of the Carina young moving group that was observed by Transiting Exoplanet Survey Satellite over two 1-yr intervals. The combined light curve contains >300 dimming events, each lasting a few hours, and as deep as 40 per cent (0.55 magnitudes). These stochastic events are correlated with a distinct, stable 1.86-d periodic signal that could be stellar rotation. Concurrent ground-based, multiband photometry show reddening consistent with interstellar medium-like dust. The star's excess emission in the infrared and emission lines in optical and infrared spectra reveal a T Tauri-like accretion disc around the star. We confirm membership of 2M0632 in the Carina group by a Bayesian analysis of its Galactic space motion and position. We combine stellar evolution models with Gaia photometry and constraints on T eff, luminosity, and the absence of detectable lithium in the photosphere to constrain the age of the group and 2M0632 to 40–60 Myr, consistent with earlier estimates. 2M0632 joins a handful of long-lived discs which challenge the canon that disc lifetimes are ≲10 Myr. All known examples surround M dwarfs, suggesting that lower X-ray/ultraviolet irradiation and slower photoevaporation by these stars can dramatically affect disc evolution. The multiplanet systems spawned by long-lived discs probably experienced significant orbital damping and migration into close-in, resonant orbits, and perhaps represented by the TRAPPIST-1 system. [ABSTRACT FROM AUTHOR]