Sound Source Localization (SSL) enabling technology for applications such as surveillance and robotics. While traditional Signal Processing (SP)-based SSL methods provide analytic solutions under specific signal and noise assumptions, recent Deep Learning (DL)-based methods have significantly outperformed them. However, their success depends on extensive training data and substantial computational resources. Moreover, they often rely on large-scale annotated spatial data and may struggle when adapting to evolving sound classes. To mitigate these challenges, we propose a novel Class Incremental Learning (CIL) approach, termed SSL-CIL, which avoids serious accuracy degradation due to catastrophic forgetting by incrementally updating the DL-based SSL model through a closed-form analytic solution. In particular, data privacy is ensured since the learning process does not revisit any historical data (exemplar-free), which is more suitable for smart home scenarios. Empirical results in the public SSLR dataset demonstrate the superior performance of our proposal, achieving a localization accuracy of 90.9%, surpassing other competitive methods.