We present a geometrically exact Kirchhoff-like electroelastic rod theory wherein the contribution of free space energy is also factored in. In addition to the usual mechanical variables such as the rod's centerline and cross-section orientation, three electric potential parameters are also introduced to account for the variation in electric potential within the rod's cross-section as well as along the rod length. The free space energy is included through an electric flux-like variable acting on the lateral surface of the rod. The governing equations of this rod are derived through balance of forces and moments (arising from both mechanical and electric sources) for mechanical variables, weighted integration of the three-dimensional electric displacement equation in the rod's cross-section for electric potential parameters and through the asymptotic expansion of a boundary integral equation for the electric flux variable. We also obtain the expressions of internal contact moment and conjugates of electric potential parameters in terms of the rod's aforementioned mechanical and electric variables. A specific choice of the three-dimensional electroelastic stored energy density is further taken catering to piezoelectic materials and dielectric elastomers and the rod's constitutive equations are rederived which exhibit interesting interplay between mechanical and electric variables. The presented work can help improve the design of soft electroelastic rod-like actuators and sensors.