The acquisition of rights to own or use real property by non-EEA nationals - other than for direct business use - is restricted, with the exception of acquisitions by limited liability companies in which non-residents do not own more than 20% of the capital and do not hold a majority of the total voting power.
"Article 1:
No one may acquire the right to own or use real property in Iceland, including fishing and hunting rights, water rights or other real property rights, whether by free assignation or enforcement measures, marriage, inheritance or deed of transfer, unless the following conditions are met:
1. In the case of an individual, he shall be an Icelandic citizen or have domicile in Iceland.
2. If several individuals are involved in a company, and each bears unlimited liability for the debts of the company, they shall all be Icelandic citizens or shall have been domiciled in Iceland for at least five continuous years.
3. In the case of a company in which some members bear unlimited liability, and others only limited liability, for the company's debts, all those who bear unlimited liability shall be Icelandic citizens or shall have been domiciled in Iceland for at least five continuous years.
4. In the case of a company in which none of the members bears unlimited liability for the company's debts, or an institution, the company or institution shall have its domicile and venue in Iceland and all its directors shall be Icelandic citizens or shall have been domiciled in Iceland for at least five continuous years. In the case of joint-stock companies, 4/5 of the share capital shall be owned by Icelandic citizens, and Icelandic citizens shall exercise the majority of the votes at shareholders' meetings.
[...]
The minister may grant permission to deviate from the conditions of the first paragraph:
1. in accordance with applications from natural or legal persons for which it is necessary to acquire the right of ownership, or the right of utilisation, over properties and the appurtenant property rights, for the purpose of direct utilisation in their business operations, or
2. in accordance with an application from a natural person if he or she is considered as having a strong connection with Iceland, e.g. by reason of marriage to an Icelandic citizen.
[...]
Notwithstanding the provisions of paragraphs 1-6, the permission of the Minister shall not be required in the following cases:
1. In the case of the hire of real property or the rights concerning real property, providing that the hire period or the period applying to other rights is three years or shorter, or where the right is reserved to terminate the agreement with one year's notice or less.
2. When the party involved enjoys rights in Iceland under the rules of the Agreement on the European Economic Area applying to the free movement of persons, the right of establishment, services or movement of capital or corresponding clauses in the Convention establishing the European Free Trade Association. The Minister shall set further rules specifying the types of real property covered by these rights and other matters concerning the execution of these rights.[...]" (Act on the Right of Ownership and Use of Real Property (19/1966), last amended 21 July 2020, Article 1 [unofficial translation])
Act on the Right of Ownership and Use of Real Property (19/1966), last amended 21 July 2020, Article 1 
[<a href="https://www.althingi.is/lagas/nuna/1966019.html">Act</a>],[<a href="https://www.government.is/Publications/Legislation/Lex/?newsid=353f66b8-f153-11e7-9421-005056bc4d74">Translation</a>]