Polymeric ionic liquids are emerging polyelectrolyte materials for modern electrochemical applications. In this paper, we propose a self-consistent field theory of a polymeric ionic liquid on a charged conductive electrode. Taking into account the conformational entropy of rather long polymerized cations within the Lifshitz theory and electrostatic and excluded volume interactions of ionic species within the mean-field approximation, we obtain a system of self-consistent field equations for the local electrostatic potential and average concentrations of monomeric units and counterions. We solve these equations in the linear approximation for the cases of a point-like charge and a flat infinite uniformly charged electrode immersed in a polymeric ionic liquid and derive analytical expressions for local ionic concentrations and electrostatic potential, and derive an analytical expression for the linear differential capacitance of the electric double layer. We also find a numerical solution to the self-consistent field equations for two types of boundary conditions for the local polymer concentration on the electrode, corresponding to the cases of the specific adsorption absence (indifferent surface) and strong short-range repulsion of the monomeric units near the charged surface (hard wall case). For both cases, we investigate the behavior of differential capacitance as a function of applied voltage for a pure polymeric ionic liquid and a polymeric ionic liquid dissolved in a polar organic solvent. We observe that the differential capacitance profile shape is strongly sensitive to the adopted boundary condition for the local polymer concentration on the electrode.