Semantic mapping with Bayesian Kernel Inference (BKI) has shown promise in providing a richer understanding of environments by effectively leveraging local spatial information. However, existing methods face challenges in constructing accurate semantic maps or reliable uncertainty maps in perceptually challenging environments due to unreliable semantic predictions. To address this issue, we propose an evidential semantic mapping framework, which integrates the evidential reasoning of Dempster-Shafer Theory of Evidence (DST) into the entire mapping pipeline by adopting Evidential Deep Learning (EDL) and Dempster's rule of combination. Additionally, the extended belief is devised to incorporate local spatial information based on their uncertainty during the mapping process. Comprehensive experiments across various off-road datasets demonstrate that our framework enhances the reliability of uncertainty maps, consistently outperforming existing methods in scenes with high perceptual uncertainties while showing semantic accuracy comparable to the best-performing semantic mapping techniques.