Galaxies are surrounded by large reservoirs of gas, mostly hydrogen, that are fed by inflows from the intergalactic medium and by outflows from galactic winds. Absorption-line measurements along the lines of sight to bright and rare background quasars indicate that this circumgalactic medium extends far beyond the starlight seen in galaxies, but very little is known about its spatial distribution. The Lyman-α transition of atomic hydrogen at a wavelength of 121.6 nanometres is an important tracer of warm (about 10 kelvin) gas in and around galaxies, especially at cosmological redshifts greater than about 1.6 at which the spectral line becomes observable from the ground. Tracing cosmic hydrogen through its Lyman-α emission has been a long-standing goal of observational astrophysics , but the extremely low surface brightness of the spatially extended emission is a formidable obstacle. A new window into circumgalactic environments was recently opened by the discovery of ubiquitous extended Lyman-α emission from hydrogen around high-redshift galaxies . Such measurements were previously limited to especially favourable systems or to the use of massive statistical averaging because of the faintness of this emission. Here we report observations of low-surface-brightness Lyman-α emission surrounding faint galaxies at redshifts between 3 and 6. We find that the projected sky coverage approaches 100 per cent. The corresponding rate of incidence (the mean number of Lyman-α emitters penetrated by any arbitrary line of sight) is well above unity and similar to the incidence rate of high-column-density absorbers frequently detected in the spectra of distant quasars . This similarity suggests that most circumgalactic atomic hydrogen at these redshifts has now been detected in emission.