Numerous industrial and scientific processes are conducted under vacuum conditions for improved cleanliness and control. Monitoring the vacuum level is crucial to ensure consistent process results. Ionization gauges represent the state-of-the-art for measuring residual gas pressure at high and ultrahigh vacuum levels. Here we propose to establish an alternative pressure gauge based on measuring the gas friction experienced by a nanomechanical membrane resonator with ultralow intrinsic damping. This approach promises more compact, durable, and accurate gauges that cover a wider pressure range than conventional ionization gauges.