Deep learning techniques have been successfully applied in Synthetic Aperture Radar (SAR) target recognition in static scenarios relying on predefined datasets. However, in real-world scenarios, models must incrementally learn new information without forgetting previously learned knowledge. Models' tendency to forget old knowledge when learning new tasks, known as catastrophic forgetting, remains an open challenge. In this paper, an incremental learning framework, called IncSAR, is proposed to mitigate catastrophic forgetting in SAR target recognition. IncSAR comprises a Vision Transformer (ViT) and a custom-designed Convolutional Neural Network (CNN) in individual branches combined through a late-fusion strategy. A denoising module, utilizing the properties of Robust Principal Component Analysis (RPCA), is introduced to alleviate the speckle noise present in SAR images. Moreover, a random projection layer is employed to enhance the linear separability of features, and a Linear Discriminant Analysis (LDA) approach is proposed to decorrelate the extracted class prototypes. Experimental results on the MSTAR and OpenSARShip benchmark datasets demonstrate that IncSAR outperforms state-of-the-art approaches, leading to an improvement from $98.05\%$ to $99.63\%$ in average accuracy and from $3.05\%$ to $0.33\%$ in performance dropping rate.