A six-year-old refugee was allegedly sexually abused on Nauru and the perpetrator caught in the act by her parents, but the man is yet to be arrested or charged, the father has said.

A police investigation is underway, but the lack of any arrest or charges against the man, who is also a refugee, has angered the father and sparked concern among families in the community.

The girl’s father, an Iranian refugee living in Nauru, hit out at the island nation’s authorities over what he said was a lack of action, and questioned differing police approaches to crimes involving refugees and locals.

No one has ever been charged for an assault against a refugee or asylum seeker on Nauru, despite multiple allegations and documented incidents including abuse of children.

On the night of 29 December the girl’s father was working at a cafe when he noticed she had gone missing and her bike was on the ground. He went searching and found his daughter alone with a man who was allegedly attempting to assault her.

In interviews with Nauru police the young girl said she had been riding her bike near the cafe before going and sitting in her father’s car. The man then approached her, and told her to follow him. He then pulled his pants down and abused her, she described.

“Then he took me there he started pulling my trousers down first and then his trousers and I told him that is not the right thing to do to a child,” read her statement to police.

The girl’s father and mother got into an altercation with the man, he said. He called the landowner of the cafe, but the perpetrator fled on his motorbike.

Assisted by staff from the Connect Settlement Services (CSS), the organisation tasked to help refugees make a life in Nauru, the girl’s family went to the police station with an interpretor and met with a child protection officer. Both he and his daughter gave statements that night, and again the next morning when police said they needed more information, he said.

In a detailed description of events obtained by Guardian Australia, the father claimed he and the accused – who worked nearby – had known each other for many months, and alleged the man had repeatedly sought to be alone with his daughter. Both parents had told the man to stay away from the girl.

The girl was taken to the hospital and later informed the man had been arrested, however the next day he was free and staying with the landlord of the cafe where the girl’s father worked “for protection”, her father said.

The father said he asked police why the man was not in custody and was told “the previous night’s statements [were] not detailed enough for the court and the case would not succeed”, adding the police officer then told him she was tasked with refugee cases. The father said he queried if this meant there were “different rules” for refugees, but said she did not answer.

He has since had a number of run-ins with the accused man on the tiny island. “Police didn’t arrest him, not the first time, not any time, and now he is free,” he said.

Guardian Australia was unable to reach the Nauruan police, but Vinci Clodumar, the president of the Nauru Law Society and who has been speaking with the girl’s father, said he was not aware of a split in investigative duties.

A week on from the alleged assault, the father told Guardian Australia he was given conflicting reasons for why no charges had been laid, claiming he was alternately told there was not enough evidence, and the man would be arrested after the holiday period.

As a result of the incident, the girl’s father said the landowner had refused to allow him to operate his restaurant, believing refugees were too dangerous.

The girl and her parents had previously spent two years in detention, with six months on Christmas Island and the final 18 months on Nauru, before being granted refugee status. About 800 people live in the Nauru community on temporary visas.

The girl’s father said his daughter still suffered mental health issues after her time in detention, exacerbated by the alleged attack. “She’s crying all the time. She gets up in the midnight,” he said. “They didn’t do anything – the child protection office.”

The young girl is also blaming her herself for the closure of the family restaurant and has displayed feelings of shame, the family said.

The father said a number of the cafe customers worked for CSS, the organisation assisting refugee resettlement in the country.

No employees are believed to have witnessed the incident, but a spokesman for the organisation said CSS was assisting the family.

“There is a police investigation under way, but we’re not privy to the details,” the spokesman said.

“We have staff who are supporting the family and making sure the alleged victim is safe and well and getting any counselling or whatever she needs.”

CSS is bound by Australian legislation preventing the disclosure of asylum seeker and detention-related information.

The Nauruan child protection services referred Guardian Australia to police, who could not be reached.

The government of Nauru, through its Australian-based public relations representatives, did not respond to questions.

A spokesman told Guardian Australia the Australian immigration minister, Peter Dutton, was aware of the allegation. “The matter is under investigation and is a matter for the Nauruan police force,” the spokesman said.