The line is a prevalent element in man-made environments, inherently encoding spatial structural information, thus making it a more robust choice for feature representation in practical applications. Despite its apparent advantages, previous rolling shutter bundle adjustment (RSBA) methods have only supported sparse feature points, which lack robustness, particularly in degenerate environments. In this paper, we introduce the first rolling shutter line-based bundle adjustment solution, RSL-BA. Specifically, we initially establish the rolling shutter camera line projection theory utilizing Plücker line parameterization. Subsequently, we derive a series of reprojection error formulations which are stable and efficient. Finally, we theoretically and experimentally demonstrate that our method can prevent three common degeneracies, one of which is first discovered in this paper. Extensive synthetic and real data experiments demonstrate that our method achieves efficiency and accuracy comparable to existing point-based rolling shutter bundle adjustment solutions.