Branches are only allowed from countries that allow Icelandic insurance companies to establish branches (reciprocity condition)
"Article 2: The following parties may conduct insurance business in this country:
1. Public limited companies that have been granted an operating license in this country pursuant to Art. Article 26
2. Insurance companies headquartered in another Member State that have been granted an operating license in their home state, cf. Articles 66 and 67
3. European companies according to Act no. 26/2004 who have been granted an operating license in a Member State, cf. Articles 66 and 67
4. Insurance companies with headquarters in a state other than a member state that receive a license to operate a branch in this country, cf. Article 73
[...]
Article 73 - Foreign insurance companies in Iceland with headquarters outside the European Economic Area:
Foreign insurance companies headquartered outside a Member State that have a license to conduct insurance business in their home country may obtain a license from the Financial Supervisory Authority to operate a branch in Iceland in accordance with this Act, provided that Icelandic insurance companies have no inferior rights in their home country.
Before insurance companies with headquarters outside a Member State receive an operating license in Iceland, the parties to the Agreement on the European Economic Area must consult on the granting of the license.[...]" (Act on Insurance Activities (56/2010), last amended 1.1.2020, Articles 2 and 73 [google translation])
Act on Insurance Activities (56/2010), last amended 1.1.2020, Articles 2 and 73
[<a href="https://www.althingi.is/lagas/nuna/2010056.html">Act</a>]