  We construct an effective field theory for an interacting Fermi liquid with
nonzero Berry curvature at zero temperature, called the Berry Fermi liquid. We
start with the extended phase space formalism, incorporating physical time into
the configuration space. This approach allows us to include the time dependence
of the background gauge fields ``covariantly'' into the symplectic structure.
Upon restricting to the physical hypersurface, the effective action that lives
on the coadjoint orbit becomes the minus free energy on the extended phase
space. We also derive the action perturbatively in external fields using the
canonical variables. For applications, we compute both linear and nonlinear
electrical responses using the Kubo formula, and identify contributions from
the electric and magnetic dipole moments, which stem from interactions breaking
parity and time-reversal symmetry. The anomalous Hall effect is confirmed using
the kinetic theory.
