A burned body was brought to a local hospital, and there was information about a second death, as yet unconfirmed, police said. The protesters were demanding to be transferred to the Greek mainland.
"The situation is tense," Lesbos Mayor Stratis Kytelis told The Associated Press. "There is information about a dead mother and her child. We haven't been able to confirm that yet."
“We have learned with deep sadness that the lives of a woman and a child were lost in a fire on #Lesvos today,” tweeted UNHCR Greece. “We are ready to support those affected and authorities by all means possible.”
A police spokesman told AP that the migrants set fire at an olive grove outside the camp just before 5 p.m. and, minutes later, inside the camp. The fires were later extinguished, he said.
About 12,000 migrants – most of them Afghans – are housed in a space designed for 3,000.
Earlier Sunday, Greek police arrested a 25-year-old Moldovan trafficker after a car carrying 12 Afghan migrants ran through a red light and crashed into a vehicle coming from a side street, killing its driver.
One migrant was seriously injured and three others were slightly injured in the crash on a road between Kavala and Thessaloniki. Some of the migrants were found in the sedan's trunk.
The Associated Press contributed to this report.