Detectives investigating the killing of the British backpacker Grace Millane in New Zealand have found a shovel they believe is linked to her death.

Auckland city police said they discovered the implement in the central west Auckland area after going through CCTV footage.

Det Insp Scott Beard said: “A shovel that was the subject of a police appeal in relation to the Grace Millane investigation … was located and seized in the central west Auckland area. Police would like to thank those members of the public who called … with various sightings of similar items.”

A 26-year-old man, who cannot be named, has been charged with Millane’s murder and appeared in an Auckland court on Monday. The last confirmed sighting of Millane, 22, from Essex, was on 1 December, the day before her birthday, with a man at the Citylife hotel in Auckland.

Police initially launched a missing person inquiry after her parents became concerned when she did not reply to birthday wishes sent on 2 December.

A week later, on 8 December, police said they were treating the case as a homicide investigation, and a day later her body was found in the Waitakere Ranges, 10 metres from a road.

Police said Millane’s family were “in the process of organising to take her home”.

In a statement on Wednesday, the family said: “Grace went off to travel the world in mid-October and arrived in New Zealand on the 20th November. By the amount of pictures and messages we received, she clearly loved this country, its people and the lifestyle.

“We all hope that what has happened to Grace will not deter even one person from venturing out into the world and discovering their own OE [overseas experience].”

On Monday, New Zealand’s prime minister, Jacinda Ardern, offered an emotional apology to the family. “Your daughter should have been safe here, and she wasn’t, and I’m sorry for that,” she said. “I cannot imagine the grief of her family and what they would be experiencing and feeling right now.”