As the development of autonomous and connected vehicles advances, the complexity of modern vehicles increases, with numerous Electronic Control Units (ECUs) integrated into the system. In an in-vehicle network, these ECUs communicate with one another using an standard protocol called Controller Area Network (CAN). Securing communication among ECUs plays a vital role in maintaining the safety and security of the vehicle. This paper proposes a robust SDN-based False Data Detection and Mitigation System (FDDMS) for in-vehicle networks. Leveraging the unique capabilities of Software-Defined Networking (SDN), FDDMS is designed to monitor and detect false data injection attacks in real-time. Specifically, we focus on brake-related ECUs within an SDN-enabled in-vehicle network. First, we decode raw CAN data to create an attack model that illustrates how false data can be injected into the system. Then, FDDMS, incorporating a Long Short Term Memory (LSTM)-based detection model, is used to identify false data injection attacks. We further propose an effective variant of DeepFool attack to evaluate the model's robustness. To countermeasure the impacts of four adversarial attacks including Fast gradient descent method, Basic iterative method, DeepFool, and the DeepFool variant, we further enhance a re-training technique method with a threshold based selection strategy. Finally, a mitigation scheme is implemented to redirect attack traffic by dynamically updating flow rules through SDN. Our experimental results show that the proposed FDDMS is robust against adversarial attacks and effectively detects and mitigates false data injection attacks in real-time.