  We expand on the known result that the Carroll algebra in $2+1$ dimensions
admits two non-trivial central extensions by computing the associated Lie
group, which we call extended Carroll group. The symplectic geometry associated
to this group is then computed to describe the motion of planar Carroll
elementary particles, in the free case, when coupled to an electromagnetic
field, and to a gravitational field. We compare to the motions of Carroll
particles in $3+1$ dimensions in the same conditions, and also give the
dynamics of Carroll particles with spin. In an electromagnetic background, the
planar Carroll dynamics differ from the known Carroll ones due to 2 new Casimir
invariants, and turn out to be non-trivial. The coupling to a gravitational
field leaves the dynamics trivial, however. Finally, we obtain the quantum
equation obeyed by Carroll wave functions \textit{via} geometric quantization.
