Yes, a free electron can emit a photon. According to quantum electrodynamics (QED), which is the theory that describes the electromagnetic interactions of charged particles, an electron can emit or absorb photons. This process is known as electron-photon interaction.
When an electron emits a photon, it typically occurs through a process called Bremsstrahlung radiation or braking radiation. In this process, the electron experiences acceleration or deceleration, and as a result, it emits a photon to conserve energy and momentum. This phenomenon is observed, for example, when high-energy electrons are deflected by the electric field of atomic nuclei or when electrons are accelerated or decelerated by other charged particles.
The emission of photons by free electrons is an important aspect of many physical phenomena, including the interaction of electrons with electromagnetic fields, the production of X-rays, and the radiation emitted by accelerating charged particles in particle accelerators.