The 40-year-old was arrested on Wednesday afternoon after an individual gained access to the premises of Thomas' Battersea School on Tuesday.
"The woman was arrested in the vicinity of the school and has been taken to a south London police station where she remains in custody", said the police.
Four-year-old Prince George, who is third in line to the throne, started at the private school last week.
Ludhiana: Thousands protest killings of Rohingyas in Myanmar
Jean Leiby of the United Nations Children's Fund (UNICEF) says Rohingya camps are crowded with children, who are in a fragile state.
"Police are part of the protective security arrangements for the Prince and we will continue to work closely with the school, which is responsible for building security on its site".
Around 560 children aged between four and 13 attend the exclusive school.
Take a look through Prince George's first day of school.
Kensington Palace said it was "aware of the issue but we would not comment further on security measures".
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