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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 south-west out to ∼2 R vir to include the NGC 1316 galaxy group. The H  I column density sensitivity (3 σ 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 (3 σ 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, starless H  I tails (only one of which 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 that was 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 current direction, length, and velocity gradient.