"She was airlifted to the Aoraki/Mount Cook Search and Rescue Base where she received medical treatment. Unfortunately she passed away a short time later."
Earlier, a helicopter was deployed to the scene where the woman fell.
"It's basically one of these tragic accidents we have all the time in the park," Mr Gaskin said this evening.
"Unfortunately, the room for error, the margin for error, is very small."
Mr Gaskin said all members of the climbing party, including the victim, were experienced and well-equipped.
"Weather conditions prior to the accident are understood to have been good. The other members of the climbing party have been airlifted back to Mt Cook Village and are helping Police piece together what led to the accident occurring," he added.
Police were contacting the woman's next of kin.
Canterbury Police urged anyone going into the back country over summer to carry a personal locator beacon in the event of an emergency.
According to Alpine Guides, previous alpine climbing experience was required to climb The Footstool.
The Australian woman was the second person to die in the park in less than a month.
Stephen Dowall went missing on November 26 after he went climbing with his friend, Wanaka man Rob Hawes.
At least 240 people have died climbing Mt Cook, and dozens of bodies have never been recovered.
Avalanche NZ mountain specialist and former Aoraki/Mt Cook search and rescue team leader Andrew Hobman said on average, four people died in the national park each year.
He said The Footstool was a sub-mountain of Mt Sefton, about 10km from the Mt Cook summit, but was subject to the same potentially dangerous conditions, "where a simple trip or a fall can have dire consequences."
Mr Hobman said this was a busy time of year for the park.
He said despite the dangers of the park's big peaks, the area's "beauty and wonder" were very attractive to climbers.
Mr Hobman said on an average midsummer's day, there could be thousands of people walking the park's tracks. Hundreds would be in the high alpine areas, he said, but fewer actually aiming to "summit" the peaks.
He said it was important climbers had patience, and prepared for the chance they'd have to sit out storms for a day or more.
Sydney doctor Mike Bishop, 53, and German father and son Raphael Viellehner, 58, and Johann, 27, died on the mountain last summer.
Barely two weeks earlier, Stu Haslett, 28, was climbing with a fellow member of the Aoraki Alpine Rescue Team when his ice pick came loose and caused him to fall to his death.
Permanent snow and glacier ice made up more than a third of the area of the national park's area.
The young Australian woman's death has been referred to the Coroner.
