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Federated clustering (FC) is an essential extension of centralized clustering designed for the federated setting, wherein the challenge lies in constructing a global similarity measure without the need to share private data. Conventional approaches to FC typically adopt extensions of centralized methods, like K-means and fuzzy c-means. However, these methods are susceptible to non-independent-and-identically-distributed (non-IID) data among clients, leading to suboptimal performance, particularly with high-dimensional data. In this paper, we present a novel approach to address these limitations by proposing a Privacy-Preserving Federated Deep Clustering based on Generative Adversarial Networks (GANs). Each client trains a local generative adversarial network (GAN) locally and uploads the synthetic data to the server. The server applies a deep clustering network on the synthetic data to establish $k$ cluster centroids, which are then downloaded to the clients for cluster assignment. Theoretical analysis demonstrates that the GAN-generated samples, shared among clients, inherently uphold certain privacy guarantees, safeguarding the confidentiality of individual data. Furthermore, extensive experimental evaluations showcase the effectiveness and utility of our proposed method in achieving accurate and privacy-preserving federated clustering.