  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$\sigma$. 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.
