In this paper, we proposed to investigate unsupervised anomaly detection in Synthetic Aperture Radar (SAR) images. Our approach considers anomalies as abnormal patterns that deviate from their surroundings without prior knowledge of their characteristics. This method deals with the crucial problems related to the presence of speckle, the spatial correlation structures in SAR images, and the lack of annotated data to train a detection algorithm. Our proposed method aims to address these issues through a self-supervised learning algorithm. First, we propose to mitigate the SAR speckle through the deep learning SAR2SAR algorithm. We then develop an Adversarial Autoencoder (AAE) to reconstruct anomaly-free SAR images from despeckled data taking into account potential spatial correlation structures. Finally, a change detection processing step is applied between the input and the output to detect anomalies. Experiments are performed to show the advantages of our method compared to the conventional Reed-Xiaoli algorithm, highlighting the importance of an efficient despeckling pre-processing step.