<p>In this work, we consider the focusing, mass-supercritical NLS equation with a nonlinear damping term. We show conditions on the nonlinearity exponents and the damping coefficient, under which the dynamics undergoes the formation of singularities in finite time. In particular, blow-up occurs also for focusing and dissipative nonlinearities of the same power, with a sufficiently small damping coefficient. This result provides rigorous proof of the non-regularizing effect of the nonlinear damping in this case, which was already suggested by previous numerical and formal results. We show that, under our assumptions, the self-similar blow-up structure for the focusing NLS dynamics is not destroyed by the dissipative effects. The nonlinear damping contributes as a forcing term in the equation for the perturbation around the self-similar profile. We estimate this term through a modulation analysis and a careful control of the time evolution of total momentum and energy functionals.</p>