Secure communication is one of the key applications of quantum networks. In recent years, following the demands for identity protection in classical communication protocols, the need for anonymity has also emerged for quantum networks. Here, we demonstrate that quantum physics allows parties—besides communicating securely over a network—to also keep their identities secret. We implement such an anonymous quantum conference key agreement by sharing multipartite entangled states in a quantum network. We demonstrate the protocol with four parties and establish keys in subsets of the network—different combinations of two and three parties—whilst keeping the participating parties anonymous. We additionally show that the protocol is verifiable and run multiple key generation and verification routines. Our work thus addresses one of the key challenges of networked communication: keeping the identities of the communicating parties private.