In　this　article　we　consider　a　Cauchy　problem　for　the　full　compressible　Navier–Stokes–Maxwell　system　arising　from　viscosity　plasmas.　This　system　is　quasilinear　hyperbolic–parabolic.　With　the　help　of　techniques　of　symmetrizers　and　the　smallness　of　non-constant　equilibrium　solutions,　we　establish　that　global　smooth　solutions　exist　and　converge　to　the　equilibrium　solution　as　the　time　approaches　infinity.　This　result　is　obtained　for　initial　data　close　to　the　steady-states.　As　a　byproduct,　we　obtain　the　global　stability　of　solutions　near　the　equilibrium　states　for　the　full　compressible　Navier–Stokes–Poisson　system　in　a　three-dimensional　torus.　©　2021,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　Switzerland　AG　part　of　Springer　Nature.