  We measure the small-scale clustering of the Data Release 16 extended Baryon
Oscillation Spectroscopic Survey Luminous Red Galaxy sample, corrected for
fibre-collisions using Pairwise Inverse Probability weights, which give
unbiased clustering measurements on all scales. We fit to the monopole and
quadrupole moments and to the projected correlation function over the
separation range $7-60\,h^{-1}$Mpc with a model based on the Aemulus
cosmological emulator to measure the growth rate of cosmic structure,
parameterized by $f\sigma_8$. We obtain a measurement of
$f\sigma_8(z=0.737)=0.408\pm0.038$, which is $1.4\sigma$ lower than the value
expected from 2018 Planck data for a flat $\Lambda$CDM model, and is more
consistent with recent weak-lensing measurements. The level of precision
achieved is 1.7 times better than more standard measurements made using only
the large-scale modes of the same sample. We also fit to the data using the
full range of scales $0.1-60\,h^{-1}$Mpc modelled by the Aemulus cosmological
emulator and find a $4.5\sigma$ tension in the amplitude of the halo velocity
field with the Planck+$\Lambda$CDM model, driven by a mismatch on the
non-linear scales. This may not be cosmological in origin, and could be due to
a breakdown in the Halo Occupation Distribution model used in the emulator.
Finally, we perform a robust analysis of possible sources of systematics,
including the effects of redshift uncertainty and incompleteness due to target
selection that were not included in previous analyses fitting to clustering
measurements on small scales.
