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Face recognition (FR) algorithms have been proven to exhibit discriminatory behaviors against certain demographic and non-demographic groups, raising ethical and legal concerns regarding their deployment in real-world scenarios. Despite the growing number of fairness studies in FR, the fairness of face presentation attack detection (PAD) has been overlooked, mainly due to the lack of appropriately annotated data. To avoid and mitigate the potential negative impact of such behavior, it is essential to assess the fairness in face PAD and develop fair PAD models. To enable fairness analysis in face PAD, we present a Combined Attribute Annotated PAD Dataset (CAAD-PAD), offering seven human-annotated attribute labels. Then, we comprehensively analyze the fairness of PAD and its relation to the nature of the training data and the Operational Decision Threshold Assignment (ODTA) through a set of face PAD solutions. Additionally, we propose a novel metric, the Accuracy Balanced Fairness (ABF), that jointly represents both the PAD fairness and the absolute PAD performance. The experimental results pointed out that female and faces with occluding features (e.g. eyeglasses, beard, etc.) are relatively less protected than male and non-occlusion groups by all PAD solutions. To alleviate this observed unfairness, we propose a plug-and-play data augmentation method, FairSWAP, to disrupt the identity/semantic information and encourage models to mine the attack clues. The extensive experimental results indicate that FairSWAP leads to better-performing and fairer face PADs in 10 out of 12 investigated cases.