Consider a plasma in a star. Now in a plasma electrons are so excited that they can no longer be held by the electromagnetic field of the nucleus. But then when we are talking about cores or red giants or white dwarfs themselves they have the electron degeneracy pressure which is due to the potential well caused by the electromagnetic field itself. Does that mean that those electrons once again come under the influence of the electromagnetic force? But doesn't that go against the definition of a plasma?
Electron degeneracy pressure is not caused by any electromagnetic interaction. It is an "ideal" effect that would be present in any high density gas of indistinguishable, non-interacting fermions.
By constraining the fermions to have high density (with gravity in this case), you force electrons to occupy states well above zero momentum and kinetic energy, since the Pauli Exclusion Principle forbids more than one electron to occupy the same quantum state.
It is this kinetic energy, even at low temperature, that produces degeneracy pressure.