Three-dimensional Weyl materials provide a controlled setting for exploring Floquet dynamics in open quantum systems, including nonequilibrium steady states (NESS). Motivated by the desire for a strongly-coupled description, we employ holography to analyze the formation and stability of a NESS in a Weyl semi-metal induced by an external circularly polarized electric field. A time-periodic steady-state solution is constructed and its stability is determined from the spectrum of out-of-equilibrium quasinormal modes (Floquet exponents). A stable region in the drive parameter space is identified; beyond a critical curve, the Floquet exponents enter the upper half of the complex plane, leading to a superharmonic response. At sufficiently strong driving, chaotic time evolution emerges in the fully nonlinear initial-boundary value problem. The anomaly-induced response of the NESS to an external magnetic field is also computed, and the resulting behavior is related to the previously proposed chiral pumping effect.