Over the past few years, the flourishing of deep learning has strongly promoted synthetic aperture radar (SAR) automatic target recognition (ATR) advancement. Many SAR ATR methods have performed well under static environment assumptions, but in real application scenarios where target categories continue to increase over time, they are prone to catastrophic forgetting on old categories of targets. In response to this problem, this paper proposes a novel method named layer-wise representative exemplar selection-based incremental learning (LwRSIL) for SAR target recognition. Specifically, we propose a layer-wise representative exemplar selection strategy, which is capable of picking out representative exemplars covering the entire class distribution. To achieve incremental learning of the ATR model, a multi-task mixed loss is formulated to continually learn reasoning capability on new categories while recalling the knowledge of old categories. Experiments on the moving and stationary target acquisition and recognition (MSTAR) dataset demonstrate that the proposed method is competitive with some state-of-the-art incremental ATR methods.