The Vlasov-Maxwell system is a fundamental kinetic model of plasma dynamics. When one considers relativistic velocities and includes effects due to collisions with a fixed background of particles, the result is the relativistic Vlasov-Maxwell-Fokker-Planck system. The first Lorentz-invariant model of this type was recently derived by Calogero and Felix in 2010. Here, we shall discuss the first well-posedness results for global-in-time classical solutions of this system posed in a lower-dimensional setting. Our methods utilize a gain in regularity stemming from the diffusive term to arrive at smooth solutions stemming from initial data which lack even weak differentiability. Additionally, we will discuss similar methods to construct solutions for the analogous non-relativistic system.