CIA Director Gina Haspel has chosen an agency veteran and close ally to be the first woman to run the part of the agency that recruits spies overseas, gathers intelligence and engages in covert actions authorized by the White House, an agency spokesman said Friday.

Elizabeth Kimber will assume control of the Central Intelligence Agency’s Directorate of Operations, until recently a male-dominated bastion also known as the clandestine service.

The...