We consider the relativistic Vlasov–Maxwell (RVM) equations in the limit when the light velocity c goes to infinity. In this regime, the RVM system converges towards the Vlasov–Poisson system and the aim of this paper is to construct asymptotic preserving numerical schemes that are robust with respect to this limit. Our approach relies on a time splitting approach for the RVM system employing an implicit time integrator for Maxwell's equations in order to damp the higher and higher frequencies present in the numerical solution. It turns out that the choice of this implicit method is crucial as even L-stable methods can lead to numerical instabilities for large values of c. A number of numerical simulations are conducted in order to investigate the performances of our numerical scheme both in the relativistic as well as in the classical limit regime. In addition, we derive the dispersion relation of the Weibel instability for the continuous and the discretized problem.