  A new kind of "super-Efimov" states of binding energies scaling as
$\ln|E_n|\sim-e^{3n\pi/4}$ were predicted by a field theory calculation for
three fermions with resonant $p$-wave interactions in two dimensions [Phys.
Rev. Lett. \textbf{110}, 235301 (2013)]. However, the universality of these
"super-Efimov" states has not been proved independently. In this Letter, we
study the three fermion system through the hyperspherical formalism. Within the
adiabatic approximation, we find that at $p$-wave resonances, the low energy
physics of states of angular momentum $\ell=\pm1$ crucially depends on the
value of an emergent dimensionless parameter $Y$ determined by the detail of
the inter-particle potential. Only if $Y$ is exactly zero, the predicted
"super-Efimov" states exist. If $Y>0$, the scaling of the bound states changes
to $\ln|E_n|\sim-(n\pi)^2/2Y$, while there are no shallow bound states if
$Y<0$.

Quantum Gases