  We study the near-threshold molecular and collisional physics of a strong
$^{40}$K p-wave Feshbach resonance through a combination of measurements,
numerical calculations, and modeling. Dimer spectroscopy employs both
radio-frequency spin-flip association in the MHz band and resonant association
in the kHz band. Systematic uncertainty in the measured binding energy is
reduced by a model that includes both the Franck-Condon overlap amplitude and
inhomogeneous broadening. Coupled-channels calculations based on mass-scaled
$^{39}$K potentials compare well to the observed binding energies and also
reveal a low-energy p-wave shape resonance in the open channel. Contrary to
conventional expectation, we observe a nonlinear variation of the binding
energy with magnetic field, and explain how this arises from the interplay of
the closed-channel ramping state with the near-threshold shape resonance in the
open channel. We develop an analytic two-channel model that includes both
resonances as well as the dipole-dipole interactions which, we show, become
important at low energy. Using this parameterization of the energy dependence
of the scattering phase, we can classify the studied $^{40}$K resonance as
broad. Throughout the paper, we compare to the well understood s-wave case, and
discuss the significant role played by van der Waals physics. The resulting
understanding of the dimer physics of p-wave resonances provides a solid
foundation for future exploration of few- and many-body orbital physics.
