In biomedical literature, it is common for entity boundaries to not align with word boundaries. Therefore, effective identification of entity spans requires approaches capable of considering tokens that are smaller than words. We introduce a novel, subword approach for named entity recognition (NER) that uses byte-pair encodings (BPE) in combination with convolutional and recurrent neural networks to produce byte-level tags of entities. We present experimental results on several standard biomedical datasets, namely the BioCreative VI Bio-ID, JNLPBA, and GENETAG datasets. We demonstrate competitive performance while bypassing the specialized domain expertise needed to create biomedical text tokenization rules.