Segment anything model (SAM) demonstrates strong generalization ability on natural image segmentation. However, its direct adaption in medical image segmentation tasks shows significant performance drops with inferior accuracy and unstable results. It may also requires an excessive number of prompt points to obtain a reasonable accuracy. For segmenting 3D radiological CT or MRI scans, a 2D SAM model has to separately handle hundreds of 2D slices. Although quite a few studies explore adapting SAM into medical image volumes, the efficiency of 2D adaption methods is unsatisfactory and 3D adaptation methods only capable of segmenting specific organs/tumors. In this work, we propose a comprehensive and scalable 3D SAM model for whole-body CT segmentation, named CT-SAM3D. Instead of adapting SAM, we propose a 3D promptable segmentation model using a (nearly) fully labeled CT dataset. To train CT-SAM3D effectively, ensuring the model's accurate responses to higher-dimensional spatial prompts is crucial, and 3D patch-wise training is required due to GPU memory constraints. For this purpose, we propose two key technical developments: 1) a progressively and spatially aligned prompt encoding method to effectively encode click prompts in local 3D space; and 2) a cross-patch prompt learning scheme to capture more 3D spatial context, which is beneficial for reducing the editing workloads when interactively prompting on large organs. CT-SAM3D is trained and validated using a curated dataset of 1204 CT scans containing 107 whole-body anatomies, reporting significantly better quantitative performance against all previous SAM-derived models by a large margin with much fewer click prompts. Our model can handle segmenting unseen organ as well. Code, data, and our 3D interactive segmentation tool with quasi-real-time responses will be made publicly available.