We report the discovery of extremely broad 21-cm HI absorption (FWZI ~1600 km/s) detected with the Westerbork Synthesis Radio Telescope in the radio source 4C37.11 (B2 0402+379). This object has been claimed to host a super-massive binary black hole (Rodriguez et al. 2006). The main features in the absorption profile are two components, separated by ~1100 km/s. The HI absorption in 4C37.11 is unusual because it is the first case where such broad absorption is found to be centred on the systemic velocity of the host galaxy and not asymmetric and blueshifted as is seen in all other galaxies with broad HI absorption. Given the large width of the absorption, we suggest that a possible explanation for the extreme properties of the HI absorption is that it is the kinematic signature of a binary black hole. If this interpretation is correct, the combined black hole mass derived from the absorption profile is consistent with that derived from the luminosity of the spheroid. If the broad absorption is indeed due to a binary black hole, this finding confirms the importance of the gaseous component in the merging process of supermassive black holes.