  Based on the recently developed interaction renormalization for the
one-dimensional $p$-wave interaction, we study the problem of a single impurity
immersed in a highly polarized Fermi sea. They interact through a narrow
$p$-wave Feshbach resonance, so the effective range $r_{0}$ naturally appears
in the system. We use the variational approach limited to single-particle-hole
excitations on top of the unperturbed Fermi sea. The polaron exhibits two
branches of solutions, namely, the attractive and repulsive branches, with
varying scattering length across the resonance. We calculate the energy
spectrum, residue and effective mass for each of the branches. We compare the
polaronic energy with the energy of the dressed molecule, and find that the
molecular state is energetically favored when increasing the interaction
strength. The critical interaction strength for the polaron-to-molecule
transition will shift to the BCS side of the $p$-wave resonance as the
effective range increases.
