If electrons are ejected out of an atom to create a constant current in a wire, then the nuclei of the atoms that lost the electron become positively ionised, which creates a positive radial electric field. Conceptually in electrostatics theory, this field must permeate throughout and beyond the confines of the wire. However, the released electron itself is negatively ionised, which in turn creates a negative radial electric field permeating within and outside of the current carrying wire.
The combined electrostatic field created both inside and outside of the wire has potentially neutralising positive and a negative components. In the concept of electrostatics, the electron and a proton have exactly the same magnitude of charge and so the fields both inside and outside the wire should always be neutral.
However, the assumption behind this logic is that when particles with equal positive and negative charges meet, all their radial line of force inter-connect in a neutralising manner. However, if the pair can still connect to other charges in the vicinity equally strongly, the answer is the opposite and that an electrostatic field does always exist both inside and outside of the current carrying wire.
I expect that someone, somewhere must have done experiments with a single proton and multiple electrons to verify this dilemma. If so they must have created a hydrogen atom with multiple electrons filling one or more of its s,p,d,f shells, but I have not heard of such an atom being created yet!