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 K3 follows the predicted threshold scaling law that K3 is energy independent on resonance, and find consistency with the scaling law K3 ∝ (k a1D) 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 K3 ∝ V −1 L , where VL is the lattice depth. Importantly, at our attainable transverse confinements and temperatures, we see a 74-fold suppression of the on-resonant threebody 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.