Accurate and robust localization and mapping are essential components for most autonomous robots. In this paper, we propose a SLAM system for building globally consistent maps, called PIN-SLAM, that is based on an elastic and compact point-based implicit neural map representation. Taking range measurements as input, our approach alternates between in- cremental learning of the local implicit signed distance field and the pose estimation using a correspondence-free, point-to- implicit model registration to the current local map. Our implicit map is based on sparse optimizable neural points, which are inherently elastic and deformable with the global pose adjustment when closing a loop. Loops are also detected using the neural point features. Extensive experiments validate that PIN-SLAM is robust to various environments and versatile to different range sensors such as LiDAR and RGB-D cameras. PIN-SLAM achieves pose estimation accuracy better or on par with the state-of-the- art LiDAR odometry or SLAM systems and outperforms the recent neural implicit SLAM approaches while maintaining a more consistent, and highly compact implicit map that can be reconstructed as accurate and complete meshes. Finally, thanks to the voxel hashing for efficient neural points indexing and the fast implicit map-based registration without closest point association, PIN-SLAM can run at the sensor frame rate on a moderate GPU.