We elaborate a numerical method for a three-dimensional hydrodynamic multi-species plasma model described by the Euler-Maxwell equations. Our method is inspired by and extends the one-dimensional scheme from [P. Degond, F. Deluzet, and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of Computational Physics, 231 (4), pp. 1917–1946, 2012]. It can cope with large variations of the Debye length $λ_D$ and is robust in the quasi-neutral limit $λ_D→0$ thanks to its asymptotic-preserving (AP) property. The key ingredients of our approach are (i) a discretization of Maxwell’s equations based on primal and dual meshes in the spirit of discrete exterior calculus (DEC) also known as the finite integration technique (FIT), (ii) a finite volume method (FVM) for the fluid equations on the dual mesh, (iii) mixed implicit-explicit timestepping, (iv) special no-flux and contact boundary conditions at an outer cut-off boundary, and (v) additional stabilization in the non-conducting region outside the plasma domain based on direct enforcement of Gauss’ law. Numerical tests provide strong evidence confirming the AP property of the proposed method.