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Few-shot Named Entity Recognition (NER) aims to identify named entities with very little annotated data. Previous methods solve this problem based on token-wise classification, which ignores the information of entity boundaries, and inevitably the performance is affected by the massive non-entity tokens. To this end, we propose a seminal span-based prototypical network (SpanProto) that tackles few-shot NER via a two-stage approach, including span extraction and mention classification. In the span extraction stage, we transform the sequential tags into a global boundary matrix, enabling the model to focus on the explicit boundary information. For mention classification, we leverage prototypical learning to capture the semantic representations for each labeled span and make the model better adapt to novel-class entities. To further improve the model performance, we split out the false positives generated by the span extractor but not labeled in the current episode set, and then present a margin-based loss to separate them from each prototype region. Experiments over multiple benchmarks demonstrate that our model outperforms strong baselines by a large margin.