The study also warns that a similar scenario could imperil island people and animals in the coming years as the climate warms and sea level rises, making fresh water harder to access.
The research took place on St. Paul Island, a remote area of Alaska that was once part of the Bering Land Bridge that joined the Americas to Asia.
The island became isolated between 14,700 and 13,500 years ago due to sea level rise during the last deglaciation, and the land area shrank significantly. Its current size is 42 square miles (110 square kilometers).
To find out what happened to the woolly mammoths, researchers collected a sediment core from one of the few freshwater lake beds on St. Paul Island.
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By analyzing the core for signs of fungi that grow on animal dung and using radiocarbon dating, they were able to tell when mammoths disappeared.
The sediment DNA "showed the presence of mammoth DNA until 5,650 years ago, plus or minus 80 years," said the study, which described the finding as the most precise dating yet of a major extinction event.
So what happened?
Researchers think that these large beasts, similar to modern day elephants, somehow persisted for some 5,000 years after mainland populations disappeared -- likely from a combination of hunting and climate change -- but were done in by the continual shortage of fresh water.
Much like elephants, which drink some 50 gallons (200 liters) per day, wooly mammoths would have struggled during what researchers found to be an extended period of dry conditions and declining water quality.
Over the course of 2,000 years, the area grew progressively smaller and drier.
When scientists analyzed mammoth bones and teeth as well as the remains of other aquatic creatures, they found signs of progressively drier conditions leading up to the extinction event.