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Fast radio bursts (FRBs) are extragalactic radio transients of extraordinary luminosity. Studying the diverse temporal and spectral behaviour recently observed in a number of FRBs may help determine the nature of the entire class. For example, a fast spinning or highly magnetised neutron star might generate the rotation-powered acceleration required to explain the bright emission. Periodic, sub-second components, suggesting such rotation, were recently reported in one FRB, and potentially in two more. Here we report the discovery of FRB 20201020A with Apertif, an FRB showing five components regularly spaced by 0.415 ms. This sub-millisecond structure in FRB 20201020A carries important clues about the progenitor of this FRB specifically, and potentially about that of FRBs in general. We thus contrast its features to the predictions of the main FRB source models. We perform a timing analysis of the FRB 20201020A components to determine the significance of the periodicity. We compare these against the timing properties of the previously reported CHIME FRBs with sub-second quasi-periodic components, and against two Apertif bursts from repeating FRB 20180916B that show complex time-frequency structure. We find the periodicity of FRB 20201020A to be marginally significant at 2.5$σ$. Its repeating subcomponents cannot be explained as a pulsar rotation since the required spin rate of over 2 kHz exceeds the limits set by typical neutron star equations of state and observations. The fast periodicity is also in conflict with a compact object merger scenario. These quasi-periodic components could, however, be caused by equidistant emitting regions in the magnetosphere of a magnetar. The sub-millisecond spacing of the components in FRB 20201020A, the smallest observed so far in a one-off FRB, may rule out both neutron-star rotation and binary mergers as the direct source of quasi-periodic FRBs.