Interplanetary dust particles hit the surfaces of airless bodies in the Solar System, generating charged and neutral gas clouds, as well as secondary ejecta dust particles. Gravitationally bound ejecta clouds that form dust exospheres were recognized by in situ dust instruments around the icy moons of Jupiter and Saturn, but have hitherto not been observed near bodies with refractory regolith surfaces. High-altitude Apollo 15 and 17 observations of a `horizon glow' indicated a putative population of high-density small dust particles near the lunar terminators, although later orbital observations yielded upper limits on the abundance of such particles that were a factor of about 104 lower than that necessary to produce the Apollo results. Here we report observations of a permanent, asymmetric dust cloud around the Moon, caused by impacts of high-speed cometary dust particles on eccentric orbits, as opposed to particles of asteroidal origin following near-circular paths striking the Moon at lower speeds. The density of the lunar ejecta cloud increases during the annual meteor showers, especially the Geminids, because the lunar surface is exposed to the same stream of interplanetary dust particles. We expect all airless planetary objects to be immersed in similar tenuous clouds of dust.