Natalie Christopher
A large rock - which may have been dislodged by her in the fall - appeared to have crushed her.
Search teams left the body there overnight so a coroner can examine her in the morning as police refused to rule out foul play.
Police spokesman Theodoros Chronopoulos said it was too early to tell if she fell or was pushed.
He said: "At this point we don't know how she died.
"We have to await the report from the coroner, which will take two to three days, to have the answers."
Investigators are said to be probing the possibility the scientist - a keen climber - was attempting to scale a steep slope without equipment, according to local reports.
Local media also reports that homicide detectives from the Greek capital of Athens have arrived in Ikaria and have taken over the case.
The ravine is about a mile from her hotel in Kerame in the north of the island.
Her 38-year-old boyfriend remains on the island. His brother and a Spanish friend have flown out to join him.
'Blood stains on bed sheets'
Cops have told The Sun Online that blood stains found on Natalie's bed sheets have been sent to a laboratory for DNA testing.
According to reports, the scientist's boyfriend said the blotches were the result of a nose bleed.
Dr Christopher's 38-year-old boyfriend reported her missing around noon on Monday when she failed to return from a jog which reportedly began at 8am.
He says he phoned her mobile at 10am, when he woke up and found she was not at the hotel, and she answered telling him she was running in a rocky area of the island.
However, after she failed to show up, he called the cops at 12 noon, police spokesman Theodore Chronopoulos told The Sun Online.
Mr Chronopoulos said: "Her partner spoke to her when he woke up at 10am and discovered she wasn't there.
"They had a conversation and she said she was out jogging.
"At 12 (noon) when she still hadn't come back and when he started to worry, he called police."
According to local media yesterday, the Oxford-educated scientist's phone was still on, prompting her worried sister to urge people not to call her to save the battery.
However, Mr Chronopoulos told The Sun Online that there has been no signal on Natalie's phone since 5pm on Monday.
He said: "The mobile phone company confirmed that her phone was on and that it had picked up a signal until about 5pm on Monday.
"Since then there hasn't been a signal."
Natalie's partner told Protothema News yesterday that she told him she was jogging up a hill in a rocky part of the island.
He says she promised to be careful.
The unnamed man claims she told him: "I can't go fast, I'll be back slowly, don't worry."
The owner of the hotel the couple were staying at said police had sealed off the room and that Natalie's partner was staying with him at his house.
The unnamed hotelier told Greek media: "He is a very good lad, he's from Cyprus. He's lost it, he has panicked, he cannot believe it."
He described the pair as a "nice little couple" and said there was no signs of problems in the relationship.
It has emerged she made a final Facebook post just hours before she went missing, where she wrote about how "beautiful" the area was.
Writing on Sunday morning, Natalie said: "Beautiful spot for swimming and deep water solo - had so much fun here today.
"Realised later there was bolted routes here too, would love to come back again to climb. Any takers?"
She was an extreme sports enthusiast and was the face of 'Cyprus Girls Can' - a campaign to encourage local women to exercise.
Her death follows the alleged rape and murder of American scientist Suzanne Eaton in Crete.
A local man has apparently confessed to the crime in which sexual assault appeared to have been the motive.
Dr Eaton was found dead in mid July at the bottom of a cave which was used by the Nazis to store weapons during World War II near Cania.
The 59-year-old mother-of-two was originally from California and lived in Germany with her husband, a British scientist.
And then if he spoke to her then, knowing she was out running, why would you phone the police at 12:00 ... did he even try to call her again?