The disappearance of a 20-year-old Icelandic woman, and the detention of a trawlerman from Greenland in connection with her murder, have brought together two of the world’s most sparsely populated countries “in solidarity and sadness”.

There was a remarkable public response when Icelandic police used social media on 15 January to appeal for help in finding Birna Brjánsdóttir, who was last seen buying a kebab in downtown Reykjavik after she left a nightclub about 4am the day before. Nearly 800 people volunteered to help and it became the biggest search-and-rescue operation in Iceland’s history.

Brjánsdóttir’s body was not found until 22 January, spotted on a beach 35 miles south of Reykjavik by a coastguard helicopter. On the same day two crewmen of a Greenland trawler were arrested in connection with her death. Within hours of the body’s discovery, candlelit vigils were held all around Greenland, 900 miles away. Last weekend more than 8,000 people took part in a march through the streets of Reykjavik in memory of Brjánsdóttir.

For three weeks the story, reduced to the single word “Birna”, has dominated headlines and been the biggest talking point in both countries.

On Thursday a court order allowed police to detain a 25-year-old Greenlander for a further two weeks in isolation, while a 30-year-old who had been held for a fortnight was released. On Friday Brjánsdóttir was buried by her grieving family.

Iceland, with no armed police on the streets, no army and the third-lowest murder rate in the world, is known as one of the world’s most peaceful countries. Greenland, by contrast, is beset by social problems and there are hopes that the tragic case of Birna might prompt Greenlanders to step up efforts to reduce violent crime, especially against women.

Yrsa Sigurðardóttir, the Icelandic writer whose crime fiction works have sold millions in many languages, said: “The security camera photos of Birna from when she was last seen, we’ve all been there, our sons and daughters have been there, walking down that same street after a night out.

“Somehow it rang a chord with the nation. Birna is such a compelling figure, so beautiful, so young, so happy. She’s never done anything to harm anybody.

“In addition to the hundreds of people that were searching, and the 8,000 on the walk, everybody has been waiting for the next news on Birna, on the edge of their seat. The whole country felt bad, we still do. We were all hoping she would be found alive, but no. There’s not a single person in the country who doesn’t feel bad about this.”

Murders are extremely rare in Iceland, averaging just over one per year, and many are the result of domestic disputes. The last unsolved murder was in 1968, when a taxi driver was shot. “I can’t ever remember a case such as this, with foreign involvement,” said Sigurðardóttir, a mother of two.

When evidence pointed to the trawlermen, whose vessel, the Polar Nanoq, had left Iceland within a day of Brjánsdóttir’s disappearance, the “Viking Squad” – the only armed police in Iceland – flew by helicopter across the Arctic Ocean and dropped officers on board. They found a large stash of hashish and Brjánsdóttir’s ID card. Her blood was also found in a red Kia rental car that both men had leased. The Polar Nanoq then returned to Iceland, where the two men were held in connection with her death, and a third for suspected drug smuggling.

After one of the murder suspects was released, Sigurðardóttir said: “The whole thing is a tragedy, but the fact that it’s not both of them somehow makes it better. Two people plotting together would have been more evil.

“The vigil in Greenland was a show of solidarity and sadness that it was one of their people involved. But the people of Greenland can’t be in any way guilty. What they did was more a show of respect and a great feeling of sadness. It was a very beautiful gesture from them.

“Let’s hope this is a catalyst to something happening about the treatment of women, the crime in Greenland. It would be a small light from a dark tragedy.”

In the week before Brjánsdóttir’s body was found, three Greenlandic women died in the small eastern town of Tasiilaq, one murdered and two who took their own lives. According to a report in Denmark in 2015, based on academic research, the per capita level of rape in remote Greenlandic towns is by far the highest in the world.

Greenland, an autonomous country within the Danish realm (Denmark, Faroe Isles and Greenland), has extremely high rates of violent crime, sexual abuse and suicide, with figures noticeably worse among the Inuit, who make up a large majority of the 56,000 population.

“Alcohol is behind most of the problems,” said Walter Tunowsky, the journalist who has covered the Birna story for Sermitsiaq, Greenland’s main newspaper. “It’s the same sort of thing you see among many indigenous people where the culture has almost disappeared within a very short number of years.

“The violence here is way over the European average, but not this type of killing. Usually the police walk in the door after a murder and it’s solved. I can’t remember any unsolved murder cases, and I can’t remember any case like Birna’s.”

Tunowsky wrote about plans for a candlelit vigil as soon as the idea was suggested on social media, and it became Greenland’s most read news story. Within five or six hours, hundreds of people turned out at the Icelandic consulate in Nuuk, and in other towns around Greenland. “As more news and evidence has come out about Birna, there has been more and more empathy with Iceland,” said Tunowsky.