No, I can't say it's rent driven per se as there are other factors but housing issues have gotten progressively worse last 20 years or so, same problem as everywhere else except that in our case it's amplified by having to structure an economy around the peculiarities of having world's second smallest independent currency in our wallets. That said, there definitely exist worse housing markets in the world, this is no Munich.
Regarding the currency, that Iceland is expensive, is partly due to rules such as our pension funds not being allowed to invest but 20% of their fund outside the country, a rule designed to reduce the downward pressure on the local currency and create demand for it that would otherwise exist to a smaller degree as the long-term returns of local investments have been worse than the performance of the "foreign" portfolio. Mix such rules which create "artificial" demand, actual demand for the currency from tourism, indexed mortgages (a concept so unbelievable I've added a link at the bottom) driving up prices and you end up with a situation where economic problems are one of the few things that we don't have to import.
Icelander here as well, I've worked in Norway, the UK and Germany. An 80k salary here buys you the lifestyle of a 50k salary in Berlin, the only reason I moved back is kids and grandparents.
https://www.islandsbanki.is/en/news/what-is-an-indexed-mortg...