Just so you’re aware, off roading is illegal in Iceland. There are dedicated F roads you can drive on, and those are only open seasonally. While Icelanders love to lift their trucks, most of the F roads are pretty docile compared to trails we have in the US. Im sure it would be a fun time anyway, just don’t expect the same experience you have here. Also, lava rock will tear up your truck if you screw up.
You’re likely to have just as much fun renting a super Jeep FWIW. They got some old school Land Rovers, some brand new electric Land Rovers, and lots of other cool cars there.
Lastly, just in case you haven't been to Iceland, nearly anything interesting is already accessible by their road system. It's actually quite depressing, honestly, to see such incredible natural phenomenon surrounded by pavement, pay-to-park kiosks, food trucks, and tour bus after tour bus. The Icelandic government has ruined many of their natural wonders for the sake of tourism, it's even driven Icelanders away from their homes. There is/was a great exhibit at the Reykjavik Art Museum showing how the government has demolished and staged over the gorgeous natural landmarks to make them tourist friendly. They have made sure you can enjoy all of them from the comfort of a tour bus or rental car.
If you're looking for a really good offroad adventure, the Norwegian countryside would be a much better choice.