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Federated learning (FL), an attractive and promising distributed machine learning paradigm, has sparked extensive interest in exploiting tremendous data stored on ubiquitous mobile devices. However, conventional FL suffers severely from resource heterogeneity, as clients with weak computational and communication capability may be unable to complete local training using the same local training hyper-parameters. In this paper, we propose Dap-FL, a deep deterministic policy gradient (DDPG)-assisted adaptive FL system, in which local learning rates and local training epochs are adaptively adjusted by all resource-heterogeneous clients through locally deployed DDPG-assisted adaptive hyper-parameter selection schemes. Particularly, the rationality of the proposed hyper-parameter selection scheme is confirmed through rigorous mathematical proof. Besides, due to the thoughtlessness of security consideration of adaptive FL systems in previous studies, we introduce the Paillier cryptosystem to aggregate local models in a secure and privacy-preserving manner. Rigorous analyses show that the proposed Dap-FL system could guarantee the security of clients' private local models against chosen-plaintext attacks and chosen-message attacks in a widely used honest-but-curious participants and active adversaries security model. In addition, through ingenious and extensive experiments, the proposed Dap-FL achieves higher global model prediction accuracy and faster convergence rates than conventional FL, and the comprehensiveness of the adjusted local training hyper-parameters is validated. More importantly, experimental results also show that the proposed Dap-FL achieves higher model prediction accuracy than two state-of-the-art RL-assisted FL methods, i.e., 6.03% higher than DDPG-based FL and 7.85% higher than DQN-based FL.