Instruction-following navigation is a key step toward embodied intelligence. Prior benchmarks mainly focus on semantic understanding but overlook systematically evaluating navigation agents' spatial perception and reasoning capabilities. In this work, we introduce the NavSpace benchmark, which contains six task categories and 1,228 trajectory-instruction pairs designed to probe the spatial intelligence of navigation agents. On this benchmark, we comprehensively evaluate 22 navigation agents, including state-of-the-art navigation models and multimodal large language models. The evaluation results lift the veil on spatial intelligence in embodied navigation. Furthermore, we propose SNav, a new spatially intelligent navigation model. SNav outperforms existing navigation agents on NavSpace and real robot tests, establishing a strong baseline for future work.