  We study three-body scattering at zero energy for a system consisting of two
identical bosons and one distinguishable particle. Key quantities in ultracold
mixtures are the three-body effective interaction strength and decay rate, that
are both characterized by the elastic three-body transition amplitude. For
resonant interspecies $p$-wave interactions, we find that this amplitude
diverges universally, regardless of the details of the short-range
interactions. This universality originates from simple three-body scattering
processes that are either elastic or inelastic and even occurs for three
dissimilar particles with one resonant $p$-wave interaction. This universal
behavior gives rise to a remarkable regime in which three-body elastic
scattering dominates over both two-body scattering and inelastic decay in
ultracold mixtures. When the $p$-wave interactions are tuned further away from
resonance, we find a series of trimer resonances, depending strongly on the
mass ratio. Our findings are relevant for the phase diagram and lifetime of
ultracold mixtures and motivate further experimental and theoretical
exploration of $p$-wave resonant mixtures.
