  A high-order extension is presented for the electromagnetic gyrokinetic
formulation in which the parallel canonical momentum is taken as one of phase
space coordinates. The high-order displacement vector associated with the
guiding-center transformation should be considered in the long wavelength
regime. This yields addtional terms in the gyrokinetic Hamiltonian which lead
to modifications to the gyrokinetic Poisson and Amp\`ere equations. In
addition, the high-order piece of the guiding-center transformation for the
parallel canonical momentum should be also kept in the electromagnetic model.
The high-order piece contains the Ba\~nos drift effect and further modifies the
gyrokinetic Amp\`ere equation.
