  A $3$-dimensional Riemannian manifold is called Killing submersion if it
admits a Riemannian submersion over a surface such that its fibers are the
trajectories of a complete unit Killing vector field. In this paper, we give a
characterization of proper biharmonic CMC surfaces in a Killing submersion. In
the last part, we also classify the proper biharmonic Hopf cylinders in a
Killing submersion.
