In this paper, a higher-derivative model for electrodynamics is presented in a dimensional Minkowski space-time by introducing a form factor into the kinetic term of Maxwell theory as , where is a characteristic length scale. Our calculations show that for the electrostatic potential of a point charge is finite at the position of the point charge in this higher-derivative modification of Maxwell's theory. For the explicit form of the potential and the electric field of a point charge are obtained analytically in this higher-derivative electrodynamics. According to numerical estimations, the upper bound for the characteristic length scale is , where is the electroweak length scale. Finally, it should be emphasized that for the results of this paper are compatible with the results of ordinary Maxwell theory.