The wife of the British TV doctor Michael Mosley, who has gone missing on the Greek island of Symi, has said her family “will not lose hope” as the search for her husband continues.

Mosley, known for his appearances on The One Show and This Morning, disappeared during a walk in the north-east of the island on Wednesday.

His wife, Dr Clare Bailey, said: “It has been three days since Michael left the beach to go for a walk. The longest and most unbearable days for myself and my children. The search is ongoing and our family are so incredibly grateful to the people of Symi, the Greek authorities and the British consulate who are working tirelessly to help find Michael. We will not lose hope.”

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Her statement came as new CCTV footage was shared of Mosley. The footage, published by the BBC, appears to show the 67-year-old walking under an umbrella in the village of Pedi, towards a path that traverses rocky hills.

The video is believed to be one of the last two CCTV sightings before he left Pedi, which he had walked to after leaving his wife at a beach at about 1.30pm local time on Wednesday.

By Saturday Mosley’s four children had also arrived on the island in the hope of finding their father, a health expert who popularised intermittent fasting and rose to fame advocating the 5:2 diet.

Previous search efforts had focused on the cliff trail Mosley had taken to Pedi from the beach of St Nikolas, which he had travelled to by boat with his wife and the British friends they were staying with on the island.

For reasons that remain unclear, Mosley said he preferred to return on foot to Symi’s main town where the couple have a home and where it subsequently transpired he had left his mobile phone.

A picture of Mosley, taken moments before his ascent to the path at about 1.30pm, shows him in a blue cap, sunglasses, T-shirt and shorts.

“Tomorrow rescuers will turn their attention to caves in the areas of Playia and Noulia,” deputy mayor Nikitas Gryllis said.

“But what we are reading in the foreign press, that there are a network of caves on Symi that are joined like tunnels and fill up with seawater, is wrong and they are far away from where we believe he was walking. Still, they will be checked too.”

Police had originally worked on the assumption that the TV presenter and columnist had gone missing on the mile-long trail between the beach and Pedi – perhaps after slipping and taking a fall – until the CCTV footage emerged. The images, which provided the first piece of concrete evidence that he made it to the village, were taken about 20 minutes after he left St Nikolas beach.

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The search has become one of the biggest in living memory for the island. “There are about 100 people out there looking for him,” the mayor of Symi, Eleftherios Papakaloudoukas, said. “They’re working in shifts with the aid of drones.”

While Bailey had been searching for her husband in a wooded area above the village of Pedi, search teams now believe Mosley travelled through a much sparser area on the other side of the bay, Papakaloudoukas said.

The search operation involves firefighters, divers, helicopters and a specially trained search dog.

Symi and nearby islands are under a yellow weather warning for high temperatures, and the mayor said the search dog was only able to work for an hour on Saturday morning due to the heat.

The mayor said rescuers not only faced the challenge of searing temperatures but the terrain itself. “In many parts it’s not just dangerous it’s impassable,” he said. “There’s not a tree up there, just rocks and cliffs.”

Greek police said with temperatures expected to reach “above 52 degrees”, it had been decided to send more air support in the form of an emergency helicopter.

A veteran police officer coordinating the search operation said it had “become one of the biggest operations to be mounted in pursuit of an individual”. “We’ve been looking for him from the air, land and sea. Criminal investigators have been sent in, divers have been sent in, the coastguard has been patrolling the shoreline, private boats and yachts joined them, a helicopter and drones have also been deployed and still there’s been nothing, no sign that could lead us to him.”

The mayor’s daughter, Mika Papakalodouka, said some of the island’s residents were out searching for Mosley. “It’s such a small island to get lost on. It’s so weird for us. Everybody is worried and looking for him.”

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Yiannis Tsavaris, a local pharmacist, said the island’s residents had been affected by the Briton’s disappearance. “Symi is a small place. There are only a few thousand permanent residents here and believe you me there’s not a person among us who hasn’t taken this personally. This story has saddened us all. We really want this man to be found and to be found alive. A lot of local volunteers have joined the search.”

The Guardian was told that the officers would take witness statements, knock on doors and examine “every CCTV camera” on the island.

Greek rescue officials have no plans to stop the search and say it will go on for as long as it takes. The veteran security official said: “I’ve been involved in police work for over 30 years and, yes, I’m surprised that we’re still looking for him, but a man can’t just vanish in thin air.”

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