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 and when coupled to an electromagnetic 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. Finally, we obtain the quantum equation obeyed by Carroll wave functions via geometric quantization.