Although deep learning methods have achieved great success in synthetic aperture radar automatic target recognition (SAR ATR), their accuracies decline sharply, as new classes are learned, which is known as catastrophic forgetting. The overlapping or confusion between the representations of new and old classes in the feature space is the main cause of catastrophic forgetting. In this article, the incremental class anchor clustering (ICAC) is proposed to address this issue. ICAC solves this problem from three perspectives: 1) how to learn the new classes; 2) how to enable the model to recognize and classify the old classes; and 3) how to solve the imbalance between old classes and new classes. To learn the new classes, ICAC adaptively adds new anchored class centers for new classes, and the features of each new class will be clustered around the corresponding anchored class center. To enable the model to recognize and classify the old classes, ICAC stores some exemplars for the old classes to ensure the classification ability of the old classes without losing the old class centers in the feature space. At the same time, ICAC adopts knowledge distillation to further alleviate catastrophic forgetting. To solve the imbalance between old classes and new classes, ICAC proposes a learning strategy named separable learning (SL), which computes the losses of the old and new exemplars separately and then adds the two losses to make a gradient descent. Experiments on the moving and stationary target acquisition and recognition (MSTAR) dataset and the OpenSARShip dataset demonstrate the effectiveness of this method in SAR ATR.