Deep learning-based linkage of records across different databases is becoming increasingly useful in data integration and mining applications to discover new insights from multiple sources of data. However, due to privacy and confidentiality concerns, organisations often are not willing or allowed to share their sensitive data with any external parties, thus making it challenging to build/train deep learning models for record linkage across different organizations' databases. To overcome this limitation, we propose the first deep learning-based multi-party privacy-preserving record linkage (PPRL) protocol that can be used to link sensitive databases held by multiple different organisations. In our approach, each database owner first trains a local deep learning model, which is then uploaded to a secure environment and securely aggregated to create a global model. The global model is then used by a linkage unit to distinguish unlabelled record pairs as matches and non-matches. We utilise differential privacy to achieve provable privacy protection against re-identification attacks. We evaluate the linkage quality and scalability of our approach using several large real-world databases, showing that it can achieve high linkage quality while providing sufficient privacy protection against existing attacks.