Yes, a First Lady has indeed had a baby while serving in the role.
Frances Cleveland: The First and Only First Lady to Give Birth in the White House
Frances Folsom Cleveland Preston, the wife of President Grover Cleveland, holds the unique distinction of being the first and only First Lady to give birth within the White House.
Her pregnancy occurred during her husband's non-consecutive second term. On September 9, 1893, their daughter, Esther Cleveland, was born in the executive mansion. This made Esther the first and only child of a president to be born in the White House.
This event highlights a unique moment in presidential history, as it's the sole instance of a First Lady giving birth while her husband occupied the highest office in the nation.
