"We express our gratitude to all first responders including the paramedic and trained first aiders onboard at the time of the incident, the crews of the ferries who responded to the emergency calls, the Police and Westpac Rescue."
Tamarua was aboard a Zefiro Charters Ltd boat off Rangitoto Island on Thursday evening when the incident occurred.
Police this morning confirmed another guest on the boat entered the water to help Tamarua, but was unsuccessful.
"That person was not injured," police said.
"A scene examination is ongoing and specialist workgroups are assisting with this."
Police and Maritime NZ are working together to understand the circumstances leading to Tamarua's death.
Zefiro Charters' owners, Lawrence Knight and Ben Mostert, said: "We are shocked and devastated with the events that unfolded yesterday afternoon.
"Our hearts go out to the family and friends of the deceased.
"We are working with police and Maritime New Zealand who have our full co-operation, and will be conducting our own independent investigation to determine what happened."
A social media post on a Cook Islands bereavement and memorial page described Tamarua as a beloved partner, daughter and sister to two siblings.
"With heavy hearts we the Tamarua and Toparea family sadly announce the passing of Danielle Tamarua," the post from a family member said.
An offer for family and friends to pay their respects at a family home in Papatoetoe was also made.
The post noted Tamarua's body had not yet been released to her family but "in due time Danny will be brought home" to the Papatoetoe address.
The 25-year-old trained as a chef at hospitality education provider NZMA in Grafton and had worked at various restaurants around Auckland.
