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.
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