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Data-driven predictive control of connected and automated vehicles (CAVs) has received increasing attention as it can achieve safe and optimal control without relying on explicit dynamical models. However, employing the data-driven strategy involves the collection and sharing of privacy-sensitive vehicle information, which is vulnerable to privacy leakage and might further lead to malicious activities. In this paper, we develop a privacy-preserving data-enabled predictive control scheme for CAVs in a mixed traffic environment, where human-driven vehicles and CAVs coexist. We tackle external eavesdroppers and honest-but-curious central unit eavesdroppers who wiretap the communication channel of the mixed traffic system and intend to infer the CAVs' state and input information. An affine masking-based privacy protection method is designed to conceal the true state and input signals, and an extended form of the data-enabled predictive leading cruise control under different matrix structures is derived to achieve privacy-preserving optimal control for CAVs. Numerical simulations demonstrate that the proposed scheme can protect the privacy of CAVs against attackers without affecting control performance or incurring heavy computations.