With the increasing emphasis on user privacy protection, biometric recognition based on federated learning have become the latest research hotspot. However, traditional federated learning methods cannot be directly applied to finger vein recognition, due to heterogeneity of data and open-set verification. Therefore, only a few application cases have been proposed. And these methods still have two drawbacks. (1) Uniform model results in poor performance in some clients, as the finger vein data is highly heterogeneous and non-Independently Identically Distributed (non-IID). (2) On individual client, a large amount of time is underutilized, such as the time to wait for returning model from server. To address those problems, this paper proposes a Personalized and Asynchronous Federated Learning for Finger Vein Recognition (PAFedFV) framework. PAFedFV designs personalized model aggregation method to solve the heterogeneity among non-IID data. Meanwhile, it employs an asynchronized training module for clients to utilize their waiting time. Finally, extensive experiments on six finger vein datasets are conducted. Base on these experiment results, the impact of non-IID finger vein data on performance of federated learning are analyzed, and the superiority of PAFedFV in accuracy and robustness are demonstrated.