I checked and it's true :

1. Off-Road Driving: Damaging and Illegal Off-road driving is strictly prohibited in Iceland, and those caught engaging in the activity can face hefty fines of up to 100,000 ISK per person. Off-road driving can cause irreversible damage to Iceland's fragile ecosystems, particularly the moss, which takes hundreds of years to grow. Iceland's moss plays a vital role in preventing soil erosion and retaining water and humidity, making its preservation crucial. In 2018, a group of French tourists mistakenly believed off-road driving was allowed in Iceland, causing significant damage and getting stuck, leading to public outcry.

2. F-Roads: A Legal and Exciting Alternative F-roads, or mountain roads, can only be accessed by 4x4 vehicles and are open for a limited period each year, typically from July to September. F-roads are often confused with off-road driving due to their unpaved, rugged appearance and rural settings, but they are legal to drive on. Some F-roads might require crossing rivers and navigating rough terrain, which is why only 4x4 vehicles are allowed on these roads. On maps, F-roads are marked with an "F" followed by a number, e.g., "F413". Some F-roads may not be clearly labeled in person, so always check your route carefully.

Source: https://www.northbound.is/blog/251/what-is-the-difference-between-f-roads-and-off-road-driving.