No, because current is due to the flow of charges in general. Both electrons and protons can produce a current when flowing past a point, as would numerous other charged elementary particles. We are just a lot more familiar with electrons being the charge carriers, because they are the particles that are free to flow in metals. Positive charges would add to the current value like protons, and electrons would subtract. In the plasma form of hydrogen, both would be flowing simultaneously. You'd count the proton flow rate as current A, the electron flow rate as current B, and the net current at that point would be A minus B, assuming both A and B are measured in the same direction.
If you had an example with one type of particles carrying the current, of the elementary charge each, then 6.2415*10^18 charges per second is one ampere.
You would need a nuclear reaction to knock individual protons out of place on a solar panel. Plus, you'd end up with the silicon, boron, and phosphorous changing their elemental identity if their protons escaped.
That was Millikan, who "counted" the electrons in a negative one Coulomb of charge. Coulomb is known for experimentally determining the mathematics behind the electrostatic force, long before we knew about the electron or that charge was a discrete quantity.
Do you know of any example where that is the case?
It can move both protons and electrons as long as both are equally capable of moving due to the electric field. But due to other factors coming in to play (e.g. strong nuclear force, the greater mass of protons, the crystal structure of solids), it is a lot easier for an electric field to cause this motion for electrons.