  The study of state-to-state spin-exchange collisions in the vicinity of
$p$-wave Feshbach resonances offer great opportunities to explore many-body
interactions and novel quantum phases. Here, we report the observation of a
spin-exchange collision near a $p$-wave Feshbach resonance within a mixture of
the lowest and third-lowest hyperfine states of $^6$Li atoms. The spin-exchange
interaction is observed over a range of ten gausses and produces a pair of
atoms in the second-lowest hyperfine states that are captured by a deep optical
dipole trap. We apply a coupled-channel method to calculate the scattering
properties of this system. We find that the $p$-wave resonance exhibits a low
inelastic collision rate and a broad resonance profile, which is due to the
modification by the accompanying spin-exchange collisions. These findings open
up new possibilities for the creation of long-lived, strongly interacting
$p$-wave Fermi gases.
