The sole exception to the highlighted statement, which is not in conflict with Ohm's Law, is the superconductor. You can have a current with no voltage drop as the resistance of the superconductor is really zero.
Or take a lossless inductor. Apply an ac voltage. The emf induced in it is equal and opposite to applied voltage at any instant and so the resultant voltage in the circuit is always zero at any instant but the current is not zero!