ByAdam Durbin,BBC News•Nikos Papanikolaou,BBC News•Joe Inwood,BBC News, in Rhodes
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TV presenter Michael Mosley, known for popularising the 5:2 diet, has gone missing while on holiday on the Greek island of Symi.
A search and rescue operation is under way, including a police dog and a drone searching hard to reach areas.
A helicopter has been deployed from Athens to assist efforts.
Greek Police said Mr Mosley left his wife on the beach and set off on a walk to the centre of the island on Wednesday.
His phone was found in the place he was staying with his wife, who reported him missing, a police spokesperson told BBC News.
The 67-year-old broadcaster is well known for programmes including the BBC series Trust Me, I'm A Doctor and appearances on BBC's The One Show and ITV's This Morning.
After officers on the island were unable to find Mr Mosley, they requested help from the Greek fire department in Athens. Firefighters arrived in Symi from nearby Rhodes at 14:00 local time (12:00 BST) on Thursday.
Firefighters, volunteers and police officers are looking for him alongside a police dog, while a drone has also been deployed to help the search.
Officers are also searching CCTV footage for any sign of Mr Mosley.
The rescue operation is focusing on the Pedi area of Symi after a woman saw him there on Wednesday, the island's deputy mayor Ilias Chaskas told BBC News.
He said the area is considered dangerous, but noted the woman had seen Mr Mosley on the road in a safer part of Pedi.
The island's mayor, Eleftherios Papakalodoukas, said firefighters carrying out the search told him they believed it was "impossible" Mr Mosley was still there.
"It is a very small, controlled area, full of people. So if something happened to him there, we would have found him by now," he told BBC News.
Mr Papakalodoukas said he believed it was likely Mr Mosley either "followed another path" or had fallen into the sea.
An appeal saying he was missing was posted on a local Facebook group on Wednesday, alongside a picture of Mr Mosley wearing a blue cap, polo shirt and shorts.
"Have you seen this man? He set off to walk back from [Agios Nikolaos beach] at about 13.30 and failed to make it home," it said.
It was extremely hot in Symi on Wednesday, with the National Observatory of Athens reporting temperatures reaching over 40C (104F) at 15:00 local time.
Symi is part of Greece's Dodecanese island group and sits about 12 miles (19km) north-west of Rhodes. In the 2021 census it had a population of approximately 2,600 people.
The majority of its beaches are remote and people are advised to take boats to visit them.
The area surrounding the Agios Nikolaos beach is extremely rocky and difficult to hike.
Local residents have said they are trying to understand why someone would leave there on foot and go on a hike - without their phone to navigate - in such challenging conditions, rather than taking the boat back.
With temperatures forecast to reach as high as 48C on Friday, an extreme heat warning has been issued on Symi.
The combination of the sweltering weather and rugged terrain will make the task for those searching for Mr Mosley even more challenging.
A Foreign Office spokesperson said: "We are supporting the family of a British man who is missing in Greece and are in contact with the local authorities."
Mr Mosley studied medicine in London and qualified as a doctor but for the last couple of decades he has been working as a presenter, documentary maker, journalist and author.
His TV work has garnered awards, including an Emmy for a BBC science documentary The Human Face - which examined the science behind human perception of beauty.
Mr Mosley lived with tapeworms in his digestive system for six weeks after deliberately infesting himself to make his 2014 BBC Four documentary Infested! Living With Parasites.
He also writes a column for the Daily Mail, and has made several TV programmes about diet and exercise - including Channel 4 show Michael Mosley: Who Made Britain Fat?
Mr Mosley has been an advocate for intermittent fasting diets, including through the 5:2 diet and The Fast 800 diet.
He has four children with his wife Clare Bailey Mosley, who is also a doctor, author and healthy living advocate.
The couple recently appeared at the Hay Festival where Mr Mosley presented a special edition of his BBC Radio 4 series and podcast Just One Thing.
Reacting to the "shocking news", his fellow Trust Me, I'm A Doctor co-star Dr Saleyha Ahsan said she was "praying he is found safe" and she feels "sick with worry".
Presenter Jeremy Vine, who has featured Mr Mosley on his BBC Radio 2 programme, said in social media post: "I'm praying this lovely man is found and thinking of Claire and the whole Mosley family."
On Thursday's edition of The One Show, presenter Alex Jones opened the programme by expressing concern that "our friend" had gone missing.
"Our thoughts are very much with his wife Claire and the rest of his family at this worrying time. We hope for more positive news," she added.
