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8 year oldThe woman, who had been staying in a Manhattan hotel, was returning from sightseeing when she felt a warm spot on her arm, an officer told the BBC.
She turned and saw her arm on fire and a man with a cigarette lighter. She was able to pat out the fire with her hands and did not suffer any injury.
The attack at the weekend is being investigated as a possible hate crime.
A police source confirmed to the BBC the woman is Scottish but would not confirm local reports she is a dentist from Glasgow.
The woman was left with a hole the size of a 25-cent coin (about the size of a 2p-coin) on her sleeve, police said.
They have footage of a man they suspect of the crime and are appealing for witnesses.
When news emerged of the attack, the Council on American-Islamic Relations called upon the mayor to add resources to the city's hate crime investigation unit.
"We are clearly seeing a spike in attacks on individual Muslims and Islamic institutions in New York and around the country, which should be of concern to all Americans," said CAIR-NY Executive Director Afaf Nasher.
The murder of an imam and his aide in the borough of Queens in August, both shot in the head, sparked an outpouring of grief and anger.
And last week two women who were pushing their babies in prams in a Brooklyn neighbourhood were attacked, and told they "don't belong here".
The prams were knocked over but no-one was injured.
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