Due to the geometric information in the 3D face data, the 3D face recognition methods exhibit better robustness against the physical attacks compared to the 2D recognition methods. In this paper, we propose PointSurFace, a 3D face point cloud recognition method integrated with a specially designed facial surface feature extraction (SurF) module. In PointSurFace, SurF comprehensively combines the coordinate positions, normal vectors, and angle relationships, etc., of each patch in the 3D face point cloud, to explore the explicit local geometric information of 3D faces and generate facial surface features. Subsequently, the surface features are fed into an encoder consisting of four set abstraction (SA) modules and three Inverted Residual MLP (InvResMLP) modules to extract more discriminative 3D facial features, where InvResMLP could suppress the overfitting and reduce information loss in the model training. In addition, PointSurFace utilizes channel fusion operations to integrate position information into the extracted 3D facial features, further enhancing the model performance. Experimental results in 3D facial recognition and verification across multiple datasets demonstrate that PointSurFace achieves the best 3D face recognition performance to date. For example, on Lock3DFace, it achieves the state-of-the-art recognition and verification accuracy of 90.03% and 80.01%, respectively, surpassing the previous methods by 2.02% and 3.19%. The ablation studies evaluate each module of PointSurFace, suggesting that the proposed surface feature extraction module significantly enhances the model performance.