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For the control of connected and autonomous vehicles (CAVs), most existing methods focus on model-based strategies. They require explicit knowledge of car-following dynamics of human-driven vehicles that are non-trivial to identify accurately. In this paper, instead of relying on a parametric car-following model, we introduce a data-driven non-parametric strategy, called DeeP-LCC (Data-EnablEd Predictive Leading Cruise Control), to achieve safe and optimal control of CAVs in mixed traffic. We first utilize Willems' fundamental lemma to obtain a data-centric representation of mixed traffic behavior. This is justified by rigorous analysis on controllability and observability properties of mixed traffic. We then employ a receding horizon strategy to solve a finite-horizon optimal control problem at each time step, in which input/output constraints are incorporated for collision-free guarantees. Numerical experiments validate the performance of DeeP-LCC compared to a standard predictive controller that requires an accurate model. Multiple nonlinear traffic simulations further confirm its great potential on improving traffic efficiency, driving safety, and fuel economy.