In a statement released on Saturday morning (New Zealand time), the family said it had been “an extremely difficult and emotional time for us all”.
Millane, 22, was last seen in central Auckland on Saturday night and was reported missing on Wednesday.
Her father David Millane fronted a press conference at Auckland Central Police Station on Friday after flying to New Zealand from his home town of Essex in the United Kingdom.
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A tearful Millane appealed to the public to come forward with any information, “no matter how small”.
“I would just like to take this opportunity to appeal to anybody who has seen, spoken to, or come into contact with Grace over the last few days, to come forward with any detail, no matter how small, and contact the investigation team,” he said.
In the statement issued on Saturday, Millane’s family praised Auckland police “who have been outstanding in their support and efforts in helping to find Grace”.
On Friday, Detective Inspector Scott Beard said the investigation into Millane’s disappearance was “progressing rapidly”.
“At this point we hold grave fears for her safety.”
The last confirmed sighting of Grace Millane was at 9.41pm Saturday at the CityLife Hotel in Queen St.
She was wearing a black dress and white shoes.
Beard said Millane was with a “male companion” at the hotel.
He had been identified by police and spoken to.
The man, who lives in New Zealand, remained a person of interest, Beard said.
Officers had also identified a “location of interest” – an apartment at the CityLife Hotel, which was being searched.
Police were stationed at the hotel on Friday afternoon but staff said they would not comment on the investigation.
Grace Millane was on a year-long OE after graduating from the University of Lincoln in September.
The advertising graduate and keen artist arrived in New Zealand on November 20, after travelling through South America.
Photos on her Instagram showed she travelled to Cape Reinga with friends before returning to Auckland.
This was her first time travelling abroad alone.
Described as “fun loving, outgoing and [a] family-orientated person”, Millane was never out of contact with family and friends for this long, her family said.
The Millane family are working with the Lucie Blackman Trust in the UK, which helps British nationals in crisis overseas.
Chief executive of the trust, Matthew Searle, said there had been a “phenomenal” response to its social media appeal, but urged people to keep sharing and retweeting.
Searle said they were “very pleased to see the level of resources deployed by New Zealand police” and hoped to see this continue.
“Our offices remain manned 24 hours over the weekend and if anyone has any information that might assist in the investigation please get in touch with us or New Zealand police directly.”
Grace Millane is described as a European woman with brown hair and brown eyes.
She is about 168cm in height, and 58kg.
If you see Millane or have information that could assist police contact the dedicated 0800 number – 0800 676 255.