Many important advances in planet formation theory have come from the discovery of unexpected planets. The thousands of discovered exoplanets have unveiled demographic trends, such as the bimodality of planetary radius distribution known as the radius valley. Modelling these trends can probe underlying processes, e.g. the formation environment and atmospheric evolution. Here, we report the discovery and characterisation of a four-planet system around the kinematically thick-disk M-dwarf LHS 1903 with orbital periods of 2.16, 6.23, 12.57, and 29.32 days that becomes the only known M-dwarf hosting four small, well-characterised planets spanning the radius valley. We utilise high-precision transit photometry from the Transiting Exoplanet Survey Satellite (TESS) and the CHaracterising ExOPlanets Satellite (CHEOPS) to measure the radii of LHS 1903 b, c, d, and e (1.382+/-0.046, 2.046^+0.078_-0.074, 2.500^+0.078_-0.077, and 1.732^+0.059_-0.058 R_oplus). Combined with HARPS-N radial velocity data, we determine the planetary bulk densities (1.24^+0.21_-0.19, 0.53^+0.11_-0.09, 0.38^+0.09_-0.08, and 1.11^+0.33_-0.31 rho_oplus). Our compositional analysis finds that planet b is rocky, planets c and d have extended atmospheres, and LHS 1903 e does not have a gaseous envelope. Our discovery that planet e, the longest-period well-characterised terrestrial M-dwarf planet, lacks an extended atmosphere causes tension with thermally-driven mass loss radius valley predictions, but supports a gas-depleted formation explanation. The observed broken atmospheric-mass fraction trend is at odds with current formation theory, but provides further evidence for a gas-depleted formation environment for terrestrial M-dwarf planets.