Online High-Definition (HD) map construction is a key component of autonomous driving. Recent methods rely on multi-view camera images for cost-effective HD map segmentation, but cameras lack depth information for accurate scene geometry. In contrast, LiDAR provides precise 3D measurements but lacks dense semantic cues. In this work, we propose LIE, LiDAR-only semantic map construction method that employ Knowledge Distillation (KD) to handle the lack of dense semantic and texture cues. Specifically, the teacher branch fuses student LiDAR features and the corresponding 2D intensity map tile to provide dense supervision for segmenting map elements using online distillation scheme. Experimental results show that our method outperforms all single-modality approaches, achieving 8.2% higher mIoU than the state-of-the-art camera-based model on nuScenes. LIE is robust over long ranges and under challenging weather and lighting, and efficiently adapts to Argoverse2 with only 10% fine-tuning, surpassing camera-based models trained on the full dataset. Source code will be available \href{https://iv.ee.hm.edu/lie/}{here}.