Jamaica was still being hammered by strong winds and heavy rain early Thursday as Hurricane Beryl passed its southern coast, having claimed at least one life on the island. The powerful storm, which devastated communities in the eastern Caribbean earlier in the week, was headed next to the Cayman Islands.

Beryl, which was downgraded after midnight to a Category 3 storm, has killed at least eight people in total. Virtually every building on the islands of Carriacou and Petite Martinique in Grenada, where the storm made landfall on Monday as a Category 4 hurricane, lay in ruins, with marinas and a hospital destroyed, rooftops torn away and tree trunks snapped like matchsticks across the drenched earth.

“We have to rebuild from the ground up,” said Dickon Mitchell, the prime minister of Grenada.

Beryl was expected to remain a dangerous hurricane as it passed just south of the Caymans, where forecasters said the storm surge could raise water levels by up to four feet.

In Mexico, officials warned that the country could be hit twice in the coming days. Most of the eastern coast of the Yucatán Peninsula, from Puerto Costa Maya to Cancun, were under a hurricane warning as of early Thursday. By the weekend, Beryl is expected to emerge into the Gulf of Mexico, where it could restrengthen into a hurricane.

In Jamaica, hundreds of thousands of households lost power, and several communities were flooded. Officials warned that the full extent of the damage was not yet clear. Beryl was the strongest storm to approach the island in over a decade. The last time a major hurricane passed within 70 miles of Jamaica was in 2007, and it has been even longer since one made landfall.

The first confirmed death in Jamaica because of the storm came when a woman was killed as a tree fell on her house in the western parish of Hanover, the head of the country’s disaster agency, Richard Thompson, said.

A rescue team was also searching for a 20-year-old man who had been swept away in a gully in Kingston after trying to retrieve a ball that he and friends had been playing with, according to a senior police officer, Michael Phipps.

In Grenada, officials said about 98 percent of the buildings on Carriacou and Petite Martinique, where 9,000 to 10,000 people live in total, had been damaged or destroyed, including Carriacou’s main health facility, the Princess Royal Hospital. Crops were ravaged, and fallen trees and utility poles littered the streets.

The natural environment also took a beating.

“There is literally no vegetation left anywhere on the island of Carriacou,” Mr. Mitchell said after visiting the islands. “The mangroves are totally destroyed.”

The death toll may rise as recovery and aid efforts continue. Officials have reported three deaths from the storm in Grenada, two of them in Carriacou. Another was reported in the Caribbean country of St. Vincent and the Grenadines. The Venezuelan president, Nicolás Maduro, said on Tuesday that three deaths had been reported in that country’s north.

In Jamaica, emergency teams were beginning to clear the roads blocked by fallen trees, debris and utility poles in several flood-ravaged communities as Beryl moved away from the island. Some 80 roads were affected, officials said. Many houses and businesses had lost their roofs. Recovery efforts were being affected by still-torrential rainfall and gusty winds.

Electricity was restored to some areas, and Jamaicans were trying to calculate their losses. The storm pummeled the south central parish of St. Elizabeth, called the “breadbasket” of Jamaica for its role in supplying key crops.

“We have had some major damage,” said Lenworth Fulton, who heads the largest farming group in Jamaica. “Crops such as yam, coconut, coffee, carrots have been badly affected.”

Earlier on Wednesday, Prime Minister Andrew Holness of Jamaica said nearly 500 people had taken refuge in shelters across the island.

And the hurricane had ripped away a section of the roof of Jamaica’s main airport, the Norman Manley International Airport in Kingston, which closed on Tuesday night.

Jamaica’s transport minister, Daryl Vaz, said a plan was being prepared to figure out how the airport would operate while the jet bridge roof for boarding and arrivals was being repaired. The storm was expected to approach the Cayman Islands overnight Wednesday into Thursday morning.

In the Caymans, most businesses had started closing their doors as residents lined up for last-minute purchases, enduring painfully slow commutes through dense traffic.

All hotels on the island had also activated their emergency plans, and flights had already evacuated more than 1,000 people.

Premier Juliana O’Connor-Connolly of the Caymans said the shelter-in-place order would begin on Wednesday evening.

“Let us remain calm, stay prepared, look after one another as we face this challenge together,” she said in a news briefing. “We can minimize the impact of Hurricane Beryl and protect our community if we do it together.”

But the government’s actions did not put Puspa Rumba-Marcum, 40, at ease.

“Despite what the leaders say, I’m really scared,” said Ms. Rumba-Marcum, a hairstylist originally from Nepal. “I’m not sure if Cayman is well prepared for this.”

Already, the storm has set records as the first Category 4 hurricane — and then the first Category 5 storm — to form in the Atlantic Ocean so early in the season. A recent study found that with ocean temperatures rising, hurricanes in the Atlantic have become likelier to grow into a major storm within just 24 hours.

Mr. Mitchell, Grenada’s prime minister, said that the powerful storm was a direct result of global warming, and that Grenada and countries like it were on the front line of the climate crisis.

“We are no longer prepared to accept that it’s OK for us to constantly suffer significant, clearly demonstrated loss and damage arising from climatic events and be expected to rebuild year after year while the countries that are responsible for creating this situation — and exacerbating this situation — sit idly by,” he said.