  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 67''x95'' 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.
