  Inspired by the narrow Feshbach resonance in systems with the two-body
interaction, we propose the two-channel model of three-component fermions with
the three-body interaction that takes into account the finite-range effects in
low dimensions. Within this model, the $p$-wave Efimov-like effect in the
four-body sector is predicted in fractional dimensions above 1D. The impact of
the finite-range interaction on the formation of the four-body bound states in
$d=1$ is also discussed in detail.
