  We investigate the three-body recombination rate of a Fermi gas of $^6$Li
atoms confined in quasi-1D near a $p$-wave Feshbach resonance. We confirm that
the quasi-1D loss rate constant $K_3$ follows the predicted threshold scaling
law that $K_3$ is energy independent on resonance, and find consistency with
the scaling law $K_3 \propto (k \, a_{1D})^6$ far from resonance [Mehta et al.
Phys. Rev. A 76, 022711 (2007)]. Further we develop a theory based on
Breit-Wigner analysis that describes the loss feature for intermediate fields.
Lastly we measure how the loss rate constant scales with transverse confinement
and find that $K_3 \propto V_L^{-1}$, where $V_L$ is the lattice depth.
Importantly, at our attainable transverse confinements and temperatures, we see
a 74-fold suppression of the on-resonant three-body loss rate constant in
quasi-1D compared to 3D. With significant further enhancement of the transverse
confinement, this suppression may pave the way for realizing stable $p$-wave
superfluids.

Atomic Physics