Domestic cats are in every part of the planet and therefore they are bound to be in Iceland together with feral cats unless the Icelandic government had decided to eradicate domestic cats from the country which is not the case thankfully. However, feral cats were ‘culled’.
Iceland is the same as any other country in that the citizens of the country have domestic cats and most of them live in the capital Reykjavik. From the standpoint of ornithologists they have a ‘feral cat problem’ like any other country and they have at least one cat shelter in Reykjavík managed by the Icelandic Cat Protection Society.
It appears that Icelanders don’t believe in keeping their cats indoors full-time as most domestic cats are indoor/outdoor cats. There appears to have been a cull on feral cats in the 1980s and 1990s and into 2000 when there was an operation to eradicate the remaining feral cat population. I don’t like to read about this because it means that they decided to kill the cats rather than operate TNR programs. A Google search for ‘TNR Iceland’ produced no results.
There are no naturally occurring wild cat species in Iceland. The cat is not a native species to Iceland. The domestic cat was imported on ships just like most other countries.
We don’t know how many domestic cats there are in Iceland but the entire population of the country amounts to 334,252 at 2016. Most of the inhabitants live in Reykjavík (about 200,000). If one 10th of the citizens of the country own one cat then there are around 30,000 domestic cats in the country. There will be some purebred cats but not many. There will also be the odd purebred cat breeder. I notice that there is or was a Norwegian Forest Cat breeder in the country.
The summer temperature in Reykjavík is around 15°C. At the date of this article the temperature is 4°C. Domestic cats like the warmth. Therefore Iceland is not an ideal country for the domestic cat to wander about in. I suspect that most domestic cats stay indoors voluntarily. I’m also sure that the feral cat population was naturally curbed by the temperature. You will find that feral cats thrive in warmer climates such as Australia where the authorities consider that they have a very severe feral cat problem. Neither is the UK ideal for feral cats. You don’t see them. It’s too wet and cold.