The Fugitive (1963)

Kimble responds to a newspaper ad for a driver and is hired to drive a certain Peggy Franklin and her 10-year-old daughter, Nancy, to California. But Kimble doesn't realize that Peggy has abducted Nancy from her ex-husband, a pathologist named Dean Franklin. Nancy innocently removes a rabbit from Dean's research lab, unaware that the rabbit has meningitis. When Nancy contracts the sickness, Kimble risks keeping his secret safe to save Nancy before she dies.