The MeerKAT Fornax Survey maps the distribution and kinematics of atomic neutral hydrogen gas (H i) 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 ∼ R vir and stretches out to ∼ 2R vir towards south west to include the NGC 1316 galaxy group. The H i column density sensitivity (3σ over 25 km s −1 ) ranges from 5 × 10 19 cm −2 at a resolution of ∼ 10 (∼ 1 kpc at the 20 Mpc distance of Fornax) down to ∼ 10 18 cm −2 at ∼ 1 (∼ 6 kpc), and slightly below this level at the lowest resolution of ∼ 100 (∼ 10 kpc). The H i mass sensitivity (3σ over 50 km s −1 ) is 6 × 10 5 M . The H i velocity resolution is 1.4 km s −1 . In this paper we describe the survey design and H i data processing, and we present a sample of six galaxies with long, one-sided, star-less H i 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 H i tails represent the first unambiguous evidence of ram pressure shaping the distribution of H i in the Fornax cluster. The disturbed optical morphology of all host galaxies supports the idea that the tails consist of H i initially pulled out of the galaxies' stellar body by tidal forces. Ram pressure was then able to further displace the weakly bound H i and give the tails their present direction, length and velocity gradient.