We present the detection of a bright radio burst at radio frequencies between 2.2--2.3 GHz with the NASA Deep Space Network (DSN) 70 m dish (DSS-63) in Madrid, Spain from FRB~20200120E. This repeating fast radio burst (FRB) was recently discovered by the Canadian Hydrogen Intensity Mapping Experiment Fast Radio Burst (CHIME/FRB) instrument and reported to be associated with the M81 spiral galaxy at a distance of 3.6 Mpc. The high time resolution capabilities of the recording system used in this observation, together with the small amount of scattering and intrinsic brightness of the burst, allow us to explore the burst structure in unprecedented detail. We find that the burst has a duration of roughly 30 $\mu$s and is comprised of several narrow components with typical separations of 2--3 $\mu$s. The narrowest component has a width of $\lesssim$ 100 ns, which corresponds to a light travel time size as small as 30 m, the smallest associated with an FRB to date. 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 $\times$ 10$^{\text{30}}$ erg s$^{\text{-1}}$ Hz$^{\text{-1}}$, which is a factor of $\sim$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. These results provide further support for the presence of a continuum of FRB burst luminosities.