  We measure the response of a rotating sample of superfluid $^3$He-B to
spin-down to rest in the zero-temperature limit. Deviations from perfect
cylindrical symmetry in the flow environment cause the initial response to
become turbulent. The remaining high polarization of vortices along the
rotation axis suppresses the turbulent behavior and leads to laminar late-time
response. We determine the dissipation during laminar decay at $(0.13-0.22)
T_{\mathrm{c}}$ from the precession frequency of the remnant vortex cluster. We
extract the mutual friction parameter $\alpha$ and confirm that its dependence
on temperature and pressure agrees with theoretical predictions. We find that
the zero-temperature extrapolation of $\alpha$ has pressure-independent value
$\alpha(T=0) \sim 5 \cdot 10^{-4}$, which we attribute to a process where
Kelvin waves, excited at surfaces of the container, propagate into the bulk and
enhance energy dissipation via overheating vortex core-bound fermions.
