No. It is illegal to drive off-road in Iceland. Only if you have a 4x4 car which is suitable for highland driving
you can drive on highland roads and marked paths. If you are driving one of our rental cars you are not insured for water damages, so driving over unbridged rivers is not necessarily the most clever thing to do, nor does it cover damage on the undercarriage of the car. Please note that if you are caught driving off road the car rental agency is authorized to charge you for breaking that rule.
See – Can all the hostels be reached by a normal car?