Within the recently introduced auxiliary master equation approach it is possible to address steady state properties of strongly correlated impurity models, small molecules or clusters efficiently and with high accuracy. It is particularly suited for dynamical mean field theory in the nonequilibrium as well as in the equilibrium case. The method is based on the solution of an auxiliary open quantum system, which can be made quickly equivalent to the original impurity problem. In its first

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... ntation a Krylov space method was employed. Here, we aim at extending the capabilities of the approach by adopting matrix product states for the solution of the corresponding auxiliary quantum master equation. This allows for a drastic increase in accuracy and permits us to access the Kondo regime for large values of the interaction. In particular, we investigate the nonequilibrium steady state of a single impurity Anderson model and focus on the spectral properties for temperatures T below the Kondo temperature T_K and for small bias voltages phi. For the two cases considered, with T=T_K/4 and T=T_K/10 we find a clear splitting of the Kondo resonance into a two-peak structure for phi close above T_K. In the equilibrium case (phi=0) and for T=T_K/4, the obtained spectral function essentially coincides with the one from numerical renormalization group.