An electric field needs SOME source.
It doesn't have to be a battery,
it could be some collection of trapped charges.
For instance you could load a capacitor in a DC circuit,
and if you then you switch the battery out of the circuit,
but create a "short" between the capacitor plates,
the capacitor plates will cause electrons to flow
from one plate to the other till they equilibrate.
Yes - there was a battery involved in creating this situation -
there's no battery in an AC generator, but as I said before,
an electric field cannot be an "original" source.