Classification of seizure type is a key step in the clinical process for evaluating an individual who presents with seizures. It determines the course of clinical diagnosis and treatment, and its impact stretches beyond the clinical domain to epilepsy research and the development of novel therapies. Automated identification of seizure type may facilitate understanding of the disease, and seizure detection and prediction has been the focus of recent research that has sought to exploit the benefits of machine learning and deep learning architectures. Nevertheless, there is not yet a definitive solution for automating the classification of seizure type, a task that must currently be performed by an expert epileptologist. Inspired by recent advances in neural memory networks (NMNs), we introduce a novel approach for the classification of seizure type using electrophysiological data. We first explore the performance of traditional deep learning techniques which use convolutional and recurrent neural networks, and enhance these architectures by using external memory modules with trainable neural plasticity. We show that our model achieves a state-of-the-art weighted F1 score of 0.945 for seizure type classification on the TUH EEG Seizure Corpus with the IBM TUSZ preprocessed data. This work highlights the potential of neural memory networks to support the field of epilepsy research, along with biomedical research and signal analysis more broadly.