Accurate and dense mapping in large-scale environments is essential for various robot applications. Recently, implicit neural signed distance fields (SDFs) have shown promising advances in this task. However, most existing approaches employ projective distances from range data as SDF supervision, introducing approximation errors and thus degrading the mapping quality. To address this problem, we introduce N$^{3}$-Mapping, an implicit neural mapping system featuring normal-guided neural non-projective signed distance fields. Specifically, we directly sample points along the surface normal, instead of the ray, to obtain more accurate non-projective distance values from range data. Then these distance values are used as supervision to train the implicit map. For large-scale mapping, we apply a voxel-oriented sliding window mechanism to alleviate the forgetting issue with a bounded memory footprint. Besides, considering the uneven distribution of measured point clouds, a hierarchical sampling strategy is designed to improve training efficiency. Experiments demonstrate that our method effectively mitigates SDF approximation errors and achieves state-of-the-art mapping quality compared to existing approaches.