The performance of Federated Learning (FL) hinges on the effectiveness of utilizing knowledge from distributed datasets. Traditional FL methods adopt an aggregate-then-adapt framework, where clients update local models based on a global model aggregated by the server from the previous training round. This process can cause client drift, especially with significant cross-client data heterogeneity, impacting model performance and convergence of the FL algorithm. To address these challenges, we introduce FedAF, a novel aggregation-free FL algorithm. In this framework, clients collaboratively learn condensed data by leveraging peer knowledge, the server subsequently trains the global model using the condensed data and soft labels received from the clients. FedAF inherently avoids the issue of client drift, enhances the quality of condensed data amid notable data heterogeneity, and improves the global model performance. Extensive numerical studies on several popular benchmark datasets show FedAF surpasses various state-of-the-art FL algorithms in handling label-skew and feature-skew data heterogeneity, leading to superior global model accuracy and faster convergence.