Safe autonomous navigation requires not only geometric obstacle avoidance but also semantic awareness. Different object categories demand different safety clearances, and the set of relevant categories may expand as the robot explores novel environments. Vision-language models (VLMs) offer rich semantic representations for building maps of unknown scenes, yet their confidence scores are poorly calibrated, making raw predictions unreliable for safety-critical planning. Conformal prediction (CP) can provide coverage guarantees, but existing methods assume a fixed label set and stationary data, limiting their use in open-world settings. We propose Conformal Semantic Mapping for Open-World Safe Robot Navigation (CoSMO), a semantic safety mapping framework that transforms VLM embeddings into online-calibrated coverage sets and safety-aware obstacle dilations under distribution shift. By calibrating directly in the embedding space, CoSMO avoids the score rescaling that arises in confidence calibration as the vocabulary expands. Experiments on the Matterport3D (MP3D) dataset show that CoSMO reduces semantic collisions by 61.6% relative to a geometry-only baseline, while maintaining formal coverage guarantees even under delayed expert feedback. Real-world experiments on a quadruped robot further demonstrate generalization by achieving long-run empirical coverage of 90.73% against the 90% target. Project website: https://yangming911.github.io/cosmo-page/.

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