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 (∼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 suppresses significantly charge carrier mobility reaching ∼1 cm 2 V −1 s −1 at 100 K in free-standing sample. Second, it gives rise to unusual temperature-dependent optical excitations in the mid-infrared region. Relatively small charge carrier concentrations and realistic temperatures suggest that these excitations may be observed experimentally.Two-dimensional (2D) indium selenide (InSe) is a recently discovered semiconductor receiving considerable attention because of its attractive electronic properties. Thin films InSe have been proposed to be suitable for field-effect transistor applications due to their high carrier mobilities, reported to exceed 10 3 cm 2 V −1 s −1 at room temperature [1][2][3][4]. Other interesting properties of this material include tunable band gap [5-8], fullydeveloped quantum Hall effect [2], anomalous optical response [2,5,9,10], superior flexibility [11], as well as excellent thermal [12,13] and thermoelectric [14] characteristics. These observations, along with its ambient stability [15], make low-dimensional InSe an appealing candidate for numerous practical applications [16].Single-layer (SL) InSe is a layered semiconductor with an indirect energy gap in the visible range. Its electronic structure is characterized by peculiar flat regions in the valence band [17], giving rise to a giant van Hove singularity in the hole density of states [18,19]. This feature is known as a "Mexican-hat"-like band, and has been predicted for the whole family of In 2 X 2 and Ga 2 X 2 (X=S,Se,Te) compounds [17,20]. Interestingly, this peculiar shape evolves into the conventional parabolic band with increasing material's thickness, as has been theoretically predicted for InSe [18,21] and recently confirmed by angular resolved photoemission spectroscopy [22]. This observation makes ultrathin InSe films especially attractive for further studies. The interest to flat-band materials is motivated by exotic physical properties of such systems, closely related to various many-body instabilities and strong correlation effects [23][24][25][26][27]. Recent experimental discovery of flat bands and associated many-body phenomena including superconductivity in magic-angle twisted bilayer graphene [28,29] enhances enormously the interest to the problem.Electron-phonon interaction in the case of narrow bands can be dramatically different from the conven-tional case and can play a more important role. In particular, a strong electron-phonon coupling in the magicangle twisted b...