Iceland was in mourning on Sunday after a young woman missing for eight days was found dead on a beach, in a rare crime that gripped the nation.

Police said in a statement on Sunday they were treating the case as murder, although 'currently it is not possible to determine the cause of death', the Daily Mail reported.

Iceland enjoys one of the world's lowest crime rates and homicides are extremely rare - police even patrol the streets unarmed.

But 20-year-old Birna Brjansdottir was found dead on a beach south of Reykjavik after more than 725 volunteers took part in the biggest search and rescue operation in Icelandic history, according to media.

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Two Greenlandic sailors aged 25 and 30 have been held in connection with her disappearance, which happened on January 14 after a night of drinking in Reykjavik's bars.

Video surveillance footage around 5am showed her stumbling through snowy and foggy streets by herself as she bought a kebab.

Her shoes were later found in the port of Hafnarfjordur, south of Reykjavik, not far from the dock where a Greenlandic trawler, the Polar Nanoq, was moored.

Video surveillance cameras also showed a small red car, a Kia Rio, parked near the vessel around 6am - identical to a vehicle observed near the spot where Brjansdottir was last seen.

The Polar Nanoq had lifted anchor just hours after the girl went missing but members of Iceland's elite police force, known as the Viking Squad, flew out to the ship by helicopter to question the crew.

The ship returned to Reykjavik and two sailors were taken into custody.

Traces of Bransdottir's blood were later found in the red car, which had been rented by the sailors.

A country of just 330,000 people, Iceland has registered an average of just 1.8 murders per year since 2001, according to police statistics.

The killers are often under the influence of alcohol or mentally unstable.