Space debris detection is vital to space missions and space situation awareness. Convolutional neural networks are introduced to detect space debris due to their excellent performance. However, noisy labels, caused by false alarms, exist in space debris detection, and cause ambiguous targets for the training of networks, leading to networks overfitting the noisy labels and losing the ability to detect space debris. To remedy this challenge, we introduce label-noise learning to space debris detection and propose a novel label-noise learning paradigm, termed Co-correcting, to overcome the effects of noisy labels. Co-correcting comprises two identical networks, and the predictions of these networks serve as auxiliary supervised information to mutually correct the noisy labels of their peer networks. In this manner, the effect of noisy labels can be mitigated by the mutual rectification of the two networks. Empirical experiments show that Co-correcting outperforms other state-of-the-art methods of label-noise learning, such as Co-teaching and JoCoR, in space debris detection. Even with a high label noise rate, the network trained via Co-correcting can detect space debris with high detection probability.