electromagnetism - Does classical electrodynamics have $U(1)$ symmetry? If yes, how?

Quantum electrodynamics (QED) is based on $U(1)$ symmetry. What happens to this symmetry in classical electrodynamics?

Addendum The books on classical electrodynamics such as J. D. Jackson, does not mention about $U(1)$ symmetry in the context of gauge invariance (as far as I know). Gauge invariance is simply understood, in classical electrodynamics books, as the invariance of Maxwell's equations under $A_\mu\to A_\mu+\partial_\mu\chi(x)$. There is no sign of U(1) invariance that I can discover here. On the other hand, when something like Dirac equation or Dirac field is brought into the scene then the implementation U(1) transformation is clear. But that is always discussed in quantum field theory books. It appears that it is essential to have a Dirac field to understand U(1) symmetry. So the question is whether it is possible to understand the existence of U(1) symmetry in classical electrodynamics without bringing in the Dirac field into the picture?