The 40-year-old is being held in custody after her arrest on Wednesday afternoon, the Metropolitan Police said in a statement.
It comes after an individual gained access to the premises of Thomas’s Battersea School the previous day.
A police spokesman said officers attended “immediately after the issue came to light” and that security was now being reviewed.
“We are working with the school, which is attended by His Royal Highness Prince George, to review its security arrangements after the incident,” the spokesman said.
“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.”
Four-year-old Prince George, who is third in line to the throne, started at the private school last week.
A spokeswoman for Kensington Palace, the residence of the Duke and Duchess of Cambridge, said: “We are aware of this issue but won’t comment on security matters.”
They said there were no plans to change the Prince’s school.
The young royal was at school on Tuesday and Wednesday, but is not understood to have been there when the incident occurred.
New pupils at Thomas’s start on half-days, before building up to staying for lunch.
“We don’t know precisely what the security arrangements are for the four-year-old Prince but I think it would be surprising if there wasn’t some sort of police presence at the school whenever he is there,” said Sky’s crime correspondent Martin Brunt.
Thomas’s Battersea has around 560 pupils aged from four to 13 and fees for the school start at £17,604 a year.