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The body of Rosalie Anderson-Hill was was found near Summit Rd about 11.30am on Monday. Police named her late this morning.
Police believe the 21-year-old died between October 25 and the morning of October 26. A post-mortem examination is under way.
Detective Senior Sergeant Damon Wells said her death was being treated as "unexplained" while police try and piece together a timeline of her movements.
He said it was too early to say if foul play was a factor.
A Facebook group set-up for friends and family to share memories of the woman has been inundated with messages.
"I will always miss you. You were that funny, free spirited, friend I always adored and when we were together we just clicked," one post said.
"Still really don't know what to say. Thank you for being a beautiful part of our lives, you won't be forgotten," another said.
Anderson-Hill's body was found by a member of the public near the Cattlestop Crag.
Det Snr Sgt Wells said police wanted to hear from anyone who was in the area on Sunday, or saw the woman walking in the Mt Pleasant, Lyttelton or Summit Road area overnight on Sunday through to Monday morning.
Anderson-Hill is described as having a slim build, approximately 160cm tall with dreadlocked blonde hair.
She was wearing a dressing gown with a hood, white and blue paisley-patterned long pants and black and white shoes.
"She may have had a black globe-brand longboard with her," said Wells.
"This is about 1m in length with a slight dip in the middle of the board."
Police are yet to locate her longboard and would like to hear from anyone who might have found it.