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Quasars, which are exceptionally bright objects at the centres (or nuclei) of galaxies, are thought to be produced through the accretion of gas into disks surrounding supermassive black holes(1-3). There is observational evidence at galactic and circumnuclear scales(4) that gas flows inwards towards accretion disks around black holes, and such an inflow has been measured at the scale of the dusty torus that surrounds the central accretion disk(5). At even smaller scales, inflows close to an accretion disk have been suggested to explain the results of recent modelling of the response of gaseous broad emission lines to continuum variations(6,7). However, unambiguous observations of inflows that actually reach accretion disks have been elusive. Here we report the detection of redshifted broad absorption lines of hydrogen and helium atoms in a sample of quasars. The lines show broad ranges of Doppler velocities that extend continuously from zero to redshifts as high as about 5,000 kilometres per second. We interpret this as the inward motion of gases at velocities comparable to freefall speeds close to the black hole, constraining the fastest infalling gas to within 10,000 gravitational radii of the black hole (the gravitational radius being the gravitational constant multiplied by the object mass, divided by the speed of light squared). Extensive photoionization modelling yields a characteristic radial distance of the inflow of approximately 1,000 gravitational radii, possibly overlapping with the outer accretion disk.