5G and beyond 5G low power wireless networks make Internet of Things (IoT) and Cyber-Physical Systems (CPS) applications capable of serving massive amounts of devices and machines. Due to the broadcast nature of wireless networks, it is crucial to secure the communication between these devices and machines from spoofing and interception attacks. This paper is concerned with the security of carrier frequency offset (CFO) based continuous physical layer authentication. The interaction between an attacker and a defender is modeled as a dynamic discrete leader-follower game with imperfect information. In the considered model, a legitimate user (Alice) communicates with the defender/operator (Bob) and is authorized by her CFO continuously. The attacker (Eve), by listening/eavesdropping the communication between Alice and Bob, tries to learn the CFO characteristics of Alice and aims to inject malicious packets to Bob by impersonating Alice. First, by showing that the optimal attacker strategy is a threshold policy, an optimization problem of the attacker with exponentially growing action space is reduced to a tractable integer optimization problem with a single parameter, then the corresponding defender cost is derived. Extensive simulations illustrate the characteristics of optimal strategies/utilities of the players depending on the actions, and show that the defender's optimal false positive rate causes attack success probabilities to be in the order of 0.99. The results show the importance of the parameters while finding the balance between system security and efficiency.