We present new Karl G. Jansky Very Large Array (JVLA) and Very Long Baseline Array (VLBA) observations that traced the H I in the central region of 3C 84 (Perseus A). This radio source is hosted by the bright cluster galaxy NGC 1275 in the centre of the iconic Perseus cluster. With the JVLA, we detected very broad (FWHM ∼500 km s−1) H I absorption at arcsecond resolution (∼300 pc) that is centred at the systemic velocity of NGC 1275 against the bright radio continuum, suggesting that the detected gas is very close to the supermassive black hole (SMBH). However, we did not detect any absorption in the higher-resolution VLBA data against the parsec-scale radio core and jet. Based on a comparison of the properties of the H I absorption with those of the molecular circum-nuclear disc (CND) known to be present in NGC 1275, we argue that the H I seen in absorption arises from H I in this fast-rotating CND, and that neutral atomic hydrogen is present as close as ∼20 pc from the SMBH. The radio continuum providing the background for absorption arises from non-thermal synchrotron emission from the star formation activity in the CND, whose presence has been reported by earlier VLBA studies. We did not detect any signature that the H I gas is affected by an interaction with the radio jet. Thus, at this stage of the evolution of the source, the impact of the radio jet on the gas in NGC 1275 mainly creates cavities on much larger galaxy scales. This prevents the circum-galactic gas from cooling, and it does not produce gas outflows. Overall, the properties of the CND in Perseus A present a very similar case to that of Mrk 231, suggesting that, unlike often assumed, H I absorption can arise against the radio emission from star formation in a CND and is perhaps common in radio AGN. With the JVLA, we serendipitously detected a new, faint absorbing system that is redshifted by ∼2660 km s−1, in addition to the already known high-velocity absorption system that is redshifted 2850 km s−1 with respect to NGC 1275. We identify this new system as gas that is stripped from a foreground galaxy falling into the Perseus cluster. This new absorption remains undetected with the VLBA.