This paper presents a novel scheme to efficiently compress Light Detection and Ranging~(LiDAR) point clouds, enabling high-precision 3D scene archives, and such archives pave the way for a detailed understanding of the corresponding 3D scenes. We focus on 2D range images~(RIs) as a lightweight format for representing 3D LiDAR observations. Although conventional image compression techniques can be adapted to improve compression efficiency for RIs, their practical performance is expected to be limited due to differences in bit precision and the distinct pixel value distribution characteristics between natural images and RIs. We propose a novel implicit neural representation~(INR)--based RI compression method that effectively handles floating-point valued pixels. The proposed method divides RIs into depth and mask images and compresses them using patch-wise and pixel-wise INR architectures with model pruning and quantization, respectively. Experiments on the KITTI dataset show that the proposed method outperforms existing image, point cloud, RI, and INR-based compression methods in terms of 3D reconstruction and detection quality at low bitrates and decoding latency.