Partial label learning (PLL) is an important weakly supervised learning problem, where each training instance is associated with a set of candidate labels that include both the true label and additional noisy labels. Most existing PLL methods assume the candidate noisy labels are randomly chosen, which hardly holds in real-world learning scenarios. In this paper, we consider a more realistic PLL scenario with competitive label noise that is more difficult to distinguish from the true label than the random label noise. We propose a novel Mutual Learning based PLL approach named ML-PLL to address this challenging problem. ML-PLL learns a prediction network based classifier and a class-prototype based classifier cooperatively through interactive mutual learning and label correction. Moreover, we use a transformation network to model the association relationships between the true label and candidate labels, and learn it together with the prediction network to match the observed candidate labels in the training data and enhance label correction. Extensive experiments are conducted on several benchmark PLL datasets, and the proposed ML-PLL approach demonstrates state-of-the-art performance for partial label learning.