The effectiveness of a language model is influenced by its token representations, which must encode contextual information and handle the same word form having a plurality of meanings (polysemy). Currently, none of the common language modelling architectures explicitly model polysemy. We propose a language model which not only predicts the next word, but also its sense in context. We argue that this higher prediction granularity may be useful for end tasks such as assistive writing, and allow for more a precise linking of language models with knowledge bases. We find that multi-sense language modelling requires architectures that go beyond standard language models, and here propose a localized prediction framework that decomposes the task into a word followed by a sense prediction task. To aid sense prediction, we utilise a Graph Attention Network, which encodes definitions and example uses of word senses. Overall, we find that multi-sense language modelling is a highly challenging task, and suggest that future work focus on the creation of more annotated training datasets.