Is the additional term in the canonical momentum exactly equal to the momentum of the electromagnetic field?

The canonical momentum of a particle in an electromagnetic field is given by $$textbf{P}=mtextbf{v}+qtextbf{A}$$ Is the term $q textbf{A}$ equal to the momentum of the electromagnetic field (which would be $mathbf{P}_text{field} = int epsilon_0 left(textbf{E}timestextbf{B}right) dV$ ) ?

Otherwise, where is this part of the momentum stored?

When we write the rate of change of momentum in a region in terms of the stress energy tensor, we write: $$frac{d}{d t}left(mathbf{P}_{text {mech }}+mathbf{P}_{text {Field}}right)_{mathrm{alpha}}=oint_{S} sum_{beta} T_{mathrm{alpha} mathrm{beta}} n_{mathrm{beta}} d a$$ (this is equation 6.122 from section 6.7 of Jackson’s electrodynamics book) where $textbf{P}_{text {mech }} = m dot{textbf{v}}$.

This equation does not take into account the change of momentum due to change in $qtextbf{A}$, unless this term is the momentum in the field itself.