The MeerKAT Fornax Survey maps the distribution and kinematics of atomic neutral hydrogen gas (HI) in the nearby Fornax galaxy cluster using the MeerKAT telescope. The 12 deg^2 survey footprint covers the central region of the cluster out to ~ Rvir and stretches out to ~ 2 Rvir towards south west to include the NGC 1316 galaxy group. The HI column density sensitivity (3 sigma over 25 km/s) ranges from 5e+19/cm^2 at a resolution of ~ 10"(~ 1 kpc at the 20 Mpc distance of Fornax) down to ~ 1e+18/cm^2 at ~ 1' (~ 6 kpc), and slightly below this level at the lowest resolution of ~ 100"(~ 10 kpc). The HI mass sensitivity (3 sigma over 50 km/s) is 6e+5 Msun. The HI velocity resolution is 1.4 km/s. In this paper we describe the survey design and HI data processing, and we present a sample of six galaxies with long, one-sided, star-less HI tails (of which only one was previously known) radially oriented within the cluster and with measurable internal velocity gradients. We argue that the joint properties of the HI tails represent the first unambiguous evidence of ram pressure shaping the distribution of HI in the Fornax cluster. The disturbed optical morphology of all host galaxies supports the idea that the tails consist of HI initially pulled out of the galaxies' stellar body by tidal forces. Ram pressure was then able to further displace the weakly bound HI and give the tails their present direction, length and velocity gradient.