We present H I imaging of the galaxy group IC 1459 carried out with six antennas of the Australian Square Kilometre Array Pathfinder equipped with phased-array feeds. We detect and resolve H I in 11 galaxies down to a column density of ̃1020 cm-2 inside a ̃6 deg2 field and with a resolution of ̃1 arcmin on the sky and ̃8 km s-1 in velocity. We present H I images, velocity fields and integrated spectra of all detections, and highlight the discovery of three H I clouds - two in the proximity of the galaxy IC 5270 and one close to NGC 7418. Each cloud has an H I mass of ̃109 M☉ and accounts for ̃15 per cent of the H I associated with its host galaxy. Available images at ultraviolet, optical and infrared wavelengths do not reveal any clear stellar counterpart of any of the clouds, suggesting that they are not gas-rich dwarf neighbours of IC 5270 and NGC 7418. Using Parkes data, we find evidence of additional extended, low-column-density H I emission around IC 5270, indicating that the clouds are the tip of the iceberg of a larger system of gas surrounding this galaxy. This result adds to the body of evidence on the presence of intragroup gas within the IC 1459 group. Altogether, the H I found outside galaxies in this group amounts to several times 109 M☉, at least 10 per cent of the H I contained inside galaxies. This suggests a substantial flow of gas in and out of galaxies during the several billion years of the group's evolution.