  We show that hole states in recently discovered single-layer InSe are
strongly renormalized by the coupling with acoustic phonons. The coupling is
enhanced significantly at moderate hole doping ($\sim$10$^{13}$ cm$^{-2}$) due
to hexagonal warping of the Fermi surface. While the system remains dynamically
stable, its electron-phonon spectral function exhibits sharp low-energy
resonances, leading to the formation of satellite quasiparticle states near the
Fermi energy. Such many-body renormalization is predicted to have two important
consequences. First, it significantly suppresses charge carrier mobility
reaching $\sim$1 cm$^2$V$^{-1}$s$^{-1}$ at $100$ K in a freestanding sample.
Second, it gives rise to unusual temperature-dependent optical excitations in
the midinfrared region. Relatively small charge carrier concentrations and
realistic temperatures suggest that these excitations may be observed
experimentally.
