Non-reciprocity of signal transmission enhances capacity of communication channels and protects transmission quality against possible signal instabilities, thus becoming an important component ensuring coherent information processing. However, non-reciprocal transmission requires breaking time-reversal symmetry (TRS) which poses challenges of both practical and fundamental character hindering the progress. Here we report a new scheme for achieving broadband non-reciprocity using a specially engineered hybrid microwave cavity. The TRS breaking is realized via strong coherent coupling between a selected chiral mode in the microwave cavity and a single collective spin excitation (magnon) in a ferromagnetic yttrium iron garnet (YIG) sphere. The non-reciprocity in transmission is observed spanning nearly a 0.5 GHz frequency band, which outperforms by two orders of magnitude the previously achieved bandwidths. Our findings open new directions for robust coherent information processing in a broad range of systems in both classical and quantum regimes.