We propose 3D-SLNR, a new and ultra-lightweight neural representation with outstanding performance for large-scale 3D mapping. The representation defines a global signed distance function (SDF) in near-surface space based on a set of band-limited local SDFs anchored at support points sampled from point clouds. These SDFs are parameterized only by a tiny multi-layer perceptron (MLP) with no latent features, and the state of each SDF is modulated by three learnable geometric properties: position, rotation, and scaling, which make the representation adapt to complex geometries. Then, we develop a novel parallel algorithm tailored for this unordered representation to efficiently detect local SDFs where each sampled point is located, allowing for real-time updates of local SDF states during training. Additionally, a prune-and-expand strategy is introduced to enhance adaptability further. The synergy of our low-parameter model and its adaptive capabilities results in an extremely compact representation with excellent expressiveness. Extensive experiments demonstrate that our method achieves state-of-the-art reconstruction performance with less than 1/5 of the memory footprint compared with previous advanced methods.