Dorset Police said they were called at 7.16am on Wednesday about three residents who were found dead at the Gainsborough care home on Ulwell Road.
Seven people were taken to hospital as a precaution and other residents of the address were evacuated to a location where they could receive appropriate care.
Police said the deaths were “currently being treated as unexplained”, adding that detectives had confirmed “possible carbon monoxide poisoning is the main line of enquiry”.
Dorset Police said a 60-year-old local woman was arrested on suspicion of manslaughter on Wednesday night and remains in police custody.
Chief Inspector Neil Third, from the Major Crime Investigation Team, said: “As part of our investigation we have made one arrest to enable us to establish whether there were acts or omissions that amount to gross negligence and to gather as much information as we can about what happened.
Earlier on Wednesday, Dorset Constabulary Chief Constable Heather Dixey told the press there was “nothing to indicate” a danger to the wider community.
“The deaths are currently being treated as unexplained and we are continuing to liaise with our partner agencies to establish the full circumstances of what happened,” she said.
“A significant presence of various agencies, including a police cordon, will remain in the area while we work with our partners to carry out an investigation.”
“Following earlier inquiries I can confirm that potential carbon monoxide poisoning is our main line of inquiry but I would stress that there is nothing to suggest there is any risk to the general public at this time.”
“Swanage is a very close-knit community and I know these very sad deaths will have a significant impact on everyone here.”
She said the other residents were taken to a place where they could receive proper care.
“They are supported by Dorset Council and the health service,” she added.
Katie Lobban, a spokeswoman for gas distribution company SGN, said crews were called to the care home earlier on Wednesday as the South of England Gas Emergency Service.
She said engineers were being sent to complete safety checks at the building, adding: “Our thoughts are with everyone affected by this tragic incident.”
People living near the scene saw residents being taken out of the building.
One man, who did not want to be named, told the PA news agency: “They have taken all the patients out, but the emergency services are still on site.”
He said he saw some residents being taken out in wheelchairs.
He estimated there were two fire engines, three to four police cars and about eight or nine ambulances at the scene where the road remains closed.
Patricia Garnell, 53, who lives in Ulwell Road, told PA: “I knew something was going on because we started hearing sirens non-stop from about 8, 8.30am.
“While I was driving to the ferry, the road was closed. I managed to get around, got to the ferry and police, ambulances and fire crews were crossing from Poole to Swanage. It was a huge device. One I’ve never seen.