We undertake the challenging task to reveal the properties of dressed light-quarks in the space- and time-like regions. For that aim, we solved the Dyson-Schwinger equation (DSE) in Minkowski space for the quark propagator in a QCD inspired model, focusing on the realization of dynamical chiral symmetry breaking in the large coupling regime. The DSE is considered in the quenched approximation within the rainbow-ladder truncation with a massive gluon and a Pauli-Villars term, which is used to tune the IR physics of the model. The solution of the DSE in Minkowski space is performed by resorting to the integral representation of the quark self-energy and propagator, which leads to a coupled set of closed self-consistent equations for the spectral densities, taking into account finite on-mass-shell renormalization. The parameters of the model are chosen such that the gluon mass scale is consistent with recent lattice QCD calculations, the Pauli-Villars mass is lowered down to about 1 GeV to concentrate strength in the infrared momentum region, and the coupling constant and renormalized mass are tuned to reproduce lattice QCD results for the quark mass function in the Landau gauge. Future application to study the pion consistently with dynamical chiral symmetry breaking within the Bethe-Salpeter framework with self-energies in Minkowski space is also delineated.