MALMO, Sweden — A family whose members say they fled Albania to escape a blood feud was rejected for asylum by Swedish officials, who months later asked them to leave.

But so far, only one has been deported.

Sentila Kernaja, who turned 21 this week, three siblings and their disabled mother each collect 1,800 Swedish kronor, about $200, a month from the government. The government covers round-the-clock care for the mother, who is 42 and paralyzed from the waist down. The siblings attend a public high school, free of charge.

Ms. Kernaja cleans houses off the books, while inhabiting a legal limbo. “It’s ridiculous,” she said in an interview here in Malmo, Sweden’s third-largest city. “I told them to keep the money and give me a work permit.”

The family is among an estimated 18,000 people who live in Sweden even though their claims for asylum have been turned down. Their status has come under scrutiny after Rakhmat Akilov, a 39-year-old failed asylum seeker from Uzbekistan, drove a truck into a pedestrian shopping street in Stockholm on April 7 in the country’s worst terrorism attack in decades. (The toll from that attack rose to five on Friday, when a 66-year-old teacher died from her injuries.)