The behaviour of large amplitude electrostatic waves in cold plasma is investigated in the context of Born–Infeld electrodynamics. Equations of motion for a relativistic electron fluid in a fixed ion background are established using an unconstrained action principle. A simple expression for the maximum electric field of 'quasi-static' electric waves in a cold Born–Infeld plasma is deduced and its properties are analysed. A lower bound on their wavelength is established and an approximation to their frequency is determined for ultrarelativistic phase velocities.