We present the first interferometric blind HI survey of the Fornax galaxy cluster, which covers an area of 15 deg$^2$ out to the cluster $R_{vir}$. The survey has a resolution of 67x95 and 6.6 km$s^{-1}$ with a 3$sigma$ sensitivity of N(HI)~2x10$^{19}$ cm$^{-2}$ and MHI 2x10$^7$ M$_odot$. We detect 16 galaxies out of 200 spectroscopically confirmed Fornax cluster members. The detections cover ~3 orders of magnitude in HI mass, from 8x10$^6$ to 1.5x10$^{10}$ M$_odot$. 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 suggesting that they have been interacting within or on their way to Fornax. Our HI detections are HI-poorer and form stars at a lower rate than non-cluster galaxies in the same $M_star$ range. Low mass galaxies are more strongly affected throughout their infall towards the cluster. The MHI/$M_star$ ratio of Fornax galaxies is comparable to that in the Virgo cluster. At fixed $M_star$, our HI detections follow the non-cluster relation between MHI and the star formation rate, and we argue that this implies that so far they have lost their HI on a timescale $gtrsim$1-2 Gyr. Deeper inside the cluster HI removal is likely to proceed faster, as confirmed by a population of HI-undetected but H$_2$-detected star-forming galaxies. Based on ALMA data, we find a large scatter in H$_2$-to-HI mass ratio, with several galaxies showing an unusually high ratio that is probably caused by faster HI removal. We identify an HI-rich subgroup of possible interacting galaxies dominated by NGC 1365, where pre-processing is likey to have taken place.
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