We present the results of a deep survey of the nearby Sculptor group and the associated Sculptor filament taken with the Parkes 64-m radio telescope in the 21-cm emission line of neutral hydrogen. We detect 31 HI sources in the Sculptor group/filament, eight of which are new HI detections. We derive a slope of the HI mass function along the Sculptor filament of alpha = -1.10(-0.11)(+0.20), which is significantly flatter than the global mass function and consistent with the flat slopes previously found in other low-density group environments. Some physical process, such as star formation, photoionization or ram-pressure stripping, must therefore be responsible for removing neutral gas predominantly from low-mass galaxies. All of our HI detections have a confirmed or tentative optical counterpart and are likely associated with luminous rather than 'dark' galaxies. Despite a column density sensitivity of about 4 x 10(17) cm(-2), we do not find any traces of extragalactic gas or tidal streams, suggesting that the Sculptor filament is, at the current time, a relatively quiescent environment that has not seen any recent major interactions or mergers.