Using data from the H I Parkes All Sky Survey (HIPASS), we have searched for neutral hydrogen in galaxies in a region similar to25x25 deg(2) centred on NGC 1399, the nominal centre of the Fornax cluster. Within a velocity search range of 300-3700 km s(-1) and to a 3sigma lower flux limit of similar to40 mJy, 110 galaxies with H I emission were detected, one of which is previously uncatalogued. None of the detections has early-type morphology. Previously unknown velocities for 14 galaxies have been determined, with a further four velocity measurements being significantly dissimilar to published values. Identification of an optical counterpart is relatively unambiguous for more than similar to90 per cent of our H I galaxies. The galaxies appear to be embedded in a sheet at the cluster velocity which extends for more than 30degrees across the search area. At the nominal cluster distance of similar to20 Mpc, this corresponds to an elongated structure more than 10 Mpc in extent. A velocity gradient across the structure is detected, with radial velocities increasing by similar to500 km s(-1) from south-east to north-west. The clustering of galaxies evident in optical surveys is only weakly suggested in the spatial distribution of our H I detections. Of 62 H I detections within a 10degrees projected radius of the cluster centre, only two are within the core region (projected radius <1&DEG;) and less than 30 per cent are within 3.5&DEG;, suggesting a considerable deficit of H I-rich galaxies in the centre of the cluster. However, relative to the field, there is a 3(&PLUSMN;1)-fold excess of H I-rich galaxies in the outer parts of the cluster where galaxies may be infalling towards the cluster for the first time.