The existing authentication and key agreement (AKA) schemes for the internet of drones (IoD) still suffer from various security attacks and fail to ensure required security properties. Moreover, drones generally have limited memory and computation capability. Motivated by these issues, a secure and lightweight AKA protocol for IoD (SLAKA-IoD) is proposed based on physical unclonable function (PUF), “exclusive or” (XOR) operation and hash function, which are simple cryptographic operations and functions that can provide better performance. In the SLAKA-IoD protocol, a drone and the ground station (GS) perform mutual authentication and establish a secure session key between them, and any two drones can also perform mutual authentication and establish a secure session key between them. Via informal security analysis, formal security analysis using the strand space model, and security verification based on the Scyther tool, the SLAKA-IoD protocol is proven to resist various security attacks and ensure required security properties. Further comparative analysis shows that the SLAKA-IoD protocol can provide more security features, and is generally lightweight as compared with these related AKA protocols for IoD, so it is suitable for IoD.