Yes, it does, but it’s more properly known as the National and University Library of Iceland.
It’s by the far the largest library in Iceland, but despite Iceland’s long literary history, it’s a rather new addition to Reykjavík. It was only established in 1994 by an agreement between the former National Library (established in 1818) and the University Library (established in 1940). Before the library took on its current form, the National Library of Iceland was housed in the House of Collections, which now houses a museum.
Debate began already in the 1950s that it was inefficient to have two major research libraries in separate buildings. The former location in the House of Collections was also considered to be insufficient for the future needs of the National Library system, so the decision was made to construct a new home for the collection.
After some economic setbacks in the 1970s that delayed the project, the first shovel was officially put in the ground in 1978. A fun fact is that Vigdís Finnbogadóttir, Iceland’s former president and the first democratically-elected female head of state in the world, helped lay the foundation for the building in 1981. As with many such projects, it ended up being delayed and over-budget. But finally, in 1994, the 50-year anniversary of Iceland’s independence, the new building was officially opened.
Now, the National and University Library houses a collection of around 1 million items of printed material and other media formats. Important parts of the library include its legal collection, historical documents, rare books and manuscripts, academic journals, and a comprehensive collection of nearly all Icelandic-language publications.
While it also serves as the research library for the University of Iceland, because it is also the National Library, it is open to all residents of Iceland.