An efficient path planner for autonomous car-like vehicles should handle the strong kinematic constraints, particularly in confined spaces commonly encountered while maneuvering in city traffic, and should enable rapid planning, as the city traffic scenarios are highly dynamic. State-of-the-art planning algorithms handle such difficult cases at high computational cost, often yielding non-deterministic results. However, feasible local paths can be quickly generated leveraging the past planning experience gained in the same or similar environment. While learning through supervised training is problematic for real traffic scenarios, we introduce in this paper a novel neural network-based method for path planning, which employs a gradient-based self-supervised learning algorithm to predict feasible paths. This approach strongly exploits the experience gained in the past and rapidly yields feasible maneuver plans for car-like vehicles with limited steering-angle. The effectiveness of such an approach has been confirmed by computational experiments.