Within the framework of kinetic theory, the nonlinear interaction of electromagnetic waves (EMWs) with a degenerate electron-ion plasma is studied to account for the electron quantum mechanical effects. For this purpose, a specific quantum regime is considered, for which the degenerate electron Fermi velocity is assumed to be taken of the order of group velocity of EMWs. This eventually leads to the existence of nonlinear Landau damping rate for the EMWs in the presence of electron Ponderomotive force. The electrons-ion density oscillations may be arisen from the nonlinear interaction of EMWs, leading to a new type of nonlinear Schrödinger equation in terms of a complex amplitude for electromagnetic pump wave. The profiles of nonlinear damping rate reveal that EMWs become less damped for increasing the quantum tunnelling effects. The electrostatic response for the linear electrostatic waves is also investigated and derived a linear dispersion for the ion-acoustic damping rate. The latter is a direct function of electron Fermi speed and does not rely on the Bohm tunneling effect. The obtained results are numerically analyzed for the two microwaves of different harmonics in the context of nonrelativistic astrophysical dense plasma environments, e.g., white dwarfs, where the electron quantum corrections cannot be ignored.