With the increasing importance of data privacy protection, various privacy-preserving machine learning methods have been proposed. In the clustering domain, various algorithms with a federated learning framework (i.e., federated clustering) have been actively studied and showed high clustering performance while preserving data privacy. However, most of the base clusterers (i.e., clustering algorithms) used in existing federated clustering algorithms need to specify the number of clusters in advance. These algorithms, therefore, are unable to deal with data whose distributions are unknown or continually changing. To tackle this problem, this paper proposes a privacy-preserving continual federated clustering algorithm. In the proposed algorithm, an adaptive resonance theory-based clustering algorithm capable of continual learning is used as a base clusterer. Therefore, the proposed algorithm inherits the ability of continual learning. Experimental results with synthetic and real-world datasets show that the proposed algorithm has superior clustering performance to state-of-the-art federated clustering algorithms while realizing data privacy protection and continual learning ability. The source code is available at \url{https://github.com/Masuyama-lab/FCAC}.