  We present new radio observations of the binary neutron star merger GW170817
carried out with the Karl G. Jansky Very large Array (VLA) more than 3\,yrs
after the merger. Our combined dataset is derived by co-adding more than
$\approx32$\,hours of VLA time on-source, and as such provides the deepest
combined observation (rms sensitivity $\approx 0.99\,\mu$Jy) of the GW170817
field obtained to date at 3\,GHz. We find no evidence for a late-time radio
re-brightening at a mean epoch of $t\approx 1200$\,d since merger, in contrast
to a $\approx 2.1\,\sigma$ excess observed at X-ray wavelengths at the same
mean epoch. Our measurements agree with expectations from the post-peak decay
of the radio afterglow of the GW170817 structured jet. Using these results, we
constrain the parameter space of models that predict a late-time radio
re-brightening possibly arising from the high-velocity tail of the GW170817
kilonova ejecta, which would dominate the radio and X-ray emission years after
the merger (once the structured jet afterglow fades below detection level). Our
results point to a steep energy-speed distribution of the kilonova ejecta (with
energy-velocity power law index $\alpha \gtrsim 5$). We suggest possible
implications of our radio analysis, when combined with the recent tentative
evidence for a late-time re-brightening in the X-rays, and highlight the need
for continued radio-to-X-ray monitoring to test different scenarios.
