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Studies of T Tauri disks inform planet formation theory; observations of variability due to occultation by circumstellar dust are a useful probe of unresolved, planet-forming inner disks, 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 TESS over two one-year intervals. The combined light curve contains >300 dimming events, each lasting a few hours, and as deep as 40% (0.55 magnitudes). These stochastic events are correlated with a distinct, stable 1.86-day periodic signal that could be stellar rotation. Concurrent ground-based, multi-band photometry show reddening consistent with ISM-like dust. The star's excess emission in the infrared and emission lines in optical and infrared spectra, reveal a T Tauri-like accretion disk 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 Teff, 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 disks which challenge the canon that disk lifetimes are <10 Myr. All known examples surround M dwarfs, suggesting that lower X-ray/UV irradiation and slower photoevaporation by these stars can dramatically affect disk evolution. The multi-planet systems spawned by long-lived disks probably experienced significant orbital damping and migration into close-in, resonant orbits, and perhaps represented by the TRAPPIST-1 system.