We have detected a bright radio burst from FRB 20200120E with the NASA Deep Space Network (DSN) 70 m dish (DSS-63) at radio frequencies between 2.2 and 2.3 GHz. This repeating fast radio burst (FRB) is reported to be associated with a globular cluster in the M81 galactic system. With high time resolution recording, low scattering, and large intrinsic brightness of the burst, we find a burst duration of ~30 μs, comprised of several narrow components with typical separations of 2-3 μs. The narrowest component has a width of ≲100 ns, which corresponds to a light travel time size as small as 30 m. The peak flux density of the narrowest burst component is 270 Jy. We estimate the total spectral luminosity of the narrowest component of the burst to be 4 × 1030 erg s-1 Hz-1, which is a factor of ~500 above the luminosities of the so-called "nanoshots" associated with giant pulses from the Crab pulsar. This spectral luminosity is also higher than that of the radio bursts detected from the Galactic magnetar SGR 1935 + 2154 during its outburst in April 2020, but it falls on the low-end of the currently measured luminosity distribution of extragalatic FRBs, further indicating the presence of a continuum of FRB luminosities. The temporal separation of the individual components has similarities to the quasiperiodic behavior seen in the microstructure of some pulsars. The known empirical relation between the microstructure quasiperiodicity timescale and the rotation period of pulsars possibly suggests a possible pulsar as the source of this FRB, with a rotation period of a few milliseconds.