The family of a British man who died on a coronavirus-stricken cruise ship have appealed to Donald Trump and authorities in Florida to allow the vessel to dock, warning that more people on board will die unless they are allowed into port.

John Carter, 75, died on Sunday 22 March after suddenly falling ill on board the Zaandam cruise liner. His widow, 74, has spent more than a week in isolation in their cabin suffering from coronavirus symptoms, their niece, Nicola Brown, said.

John Carter, 75, who died aboard the Zaandam cruise liner. Photograph: Family photograph

Several Latin American countries refused to allow the Zaandam and its sister ship, the Rotterdam, into port. They passed through the Panama canal on Monday and both are travelling towards Fort Lauderdale, Florida, to attempt to dock. It remains unclear whether local officials will allow them into Port Everglades.

Ron DeSantis, the governor of Florida, has said the passengers, many of whom are elderly, cannot be “dumped” in his state, dismissing them as mostly “foreigners”. On Tuesday, Trump, said he would speak to DeSantis about allowing the cruise liners into port, but despite the US president’s promised intervention no local permission has yet been given.

Brown appealed to local officials in Broward county, south-east Florida, to let the Zaandam and Rotterdam into port. She said her aunt was stuck in a room with her husband’s possessions and was desperate to get off the ship.

She told the Guardian: “It has been horrendous. As soon as it happened, they had to go into isolation. She’s obviously isolated in a room with all of her husband’s items. She is not feeling well herself, she has got symptoms – a really severe cough. I have just spoken to her and she is coughing away. She is completely isolated grieving my uncle. She’s only got one channel on the TV – the news.”

Brown, an NHS midwife, said she did not know if her uncle, a businessman, had tested positive for Covid-19, but he was on a ventilator before he died. Three other people have died on the Zaandam, while nine have tested positive for Covid-19 and dozens of others are ill with flu-like symptoms.

Brown said her uncle had been feeling well on the Friday before he died but quickly deteriorated after visiting the cruise liner’s medical centre on Saturday.

Appealing to local officials and to Trump, Brown said: “These are people. They are people that are unwell and they need to get home. It is the most inhumane thing. I feel desperately sorry for the crew. I feel like the crew is being ignored and it’s all about the passengers. These are human beings and they need help, whatever nationality they are from.

“We need to allow them to dock. They cannot just leave people who are seriously ill out at sea. They have tried to put a plan in place, but they have to allow the ship to dock initially first. You cannot just leave people out at sea with nothing. More people are going to die.”

On Tuesday, local officials said they were waiting for another plan from the ships’ operator, Holland America Line, that met a series of 12 conditions, but warned that the vessels would not be allowed into the port if they failed to meet them.