In a previous work, we derived a result of semi-global minimal time robust stabilization for analytic control systems with controls entering linearly, by means of a hybrid state feedback law, under the main assumption of the absence of minimal time singular trajectories. In this paper, we investigate the Martinet case, which is a model case in $\R^3$ where singular minimizers appear, and show that such a stabilization result still holds. Namely, we prove that the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator errors.