We present the first interferometric blind HI survey of the Fornax galaxy cluster, which covers an area of 15 deg2 out to the cluster virial radius. The survey has a spatial and velocity resolution of 67″ × 95″(∼6 × 9 kpc at the Fornax cluster distance of 20 Mpc) and 6.6 km s−1 and a 3σ sensitivity of NHI ∼ 2 × 1019 cm−2 and MHI ∼ 2 × 107 M⊙, respectively. We detect 16 galaxies out of roughly 200 spectroscopically confirmed Fornax cluster members. The detections cover about three orders of magnitude in HI mass, from 8 × 106 to 1.5 × 1010 M⊙. They avoid the central, virialised region of the cluster both on the sky and in projected phase-space, showing that they are recent arrivals and that, in Fornax, HI is lost within a crossing time, ∼2 Gyr. Half of these galaxies exhibit a disturbed HI morphology, including several cases of asymmetries, tails, offsets between HI and optical centres, and a case of a truncated HI disc. This suggests that these recent arrivals have been interacting with other galaxies, the large-scale potential or the intergalactic medium, within or on their way to Fornax. As a whole, our Fornax HI detections are HI-poorer and form stars at a lower rate than non-cluster galaxies in the same M⋆ range. This is particularly evident at M⋆ ≲ 109 M⊙, indicating that low mass galaxies are more strongly affected throughout their infall towards the cluster. The MHI/M⋆ ratio of Fornax galaxies is comparable to that in the Virgo cluster. At fixed M⋆, our HI detections follow the non-cluster relation between MHI and the star formation rate, and we argue that this implies that thus far they have lost their HI on a timescale ≳1−2 Gyr. Deeper inside the cluster HI removal is likely to proceed faster, as confirmed by a population of HI-undetected but H2-detected star-forming galaxies. Overall, based on ALMA data, we find a large scatter in H2-to-HI mass ratio, with several galaxies showing an unusually high ratio that is probably caused by faster HI removal. Finally, we identify an HI-rich subgroup of possible interacting galaxies dominated by NGC 1365, where pre-processing is likely to have taken place.