MCMINNVILLE, Tenn.—Whitney Brown was in labor with her first baby when suddenly she couldn’t breathe.

Convulsions shook her body. Ms. Brown’s blood pressure and oxygen levels dropped, and the baby’s heart rate plunged. Nurses at Saint Thomas River Park Hospital called obstetrician Dawnmarie Riley, who minutes later burst into the operating room in such a rush her hospital scrubs were inside out.

Dr. Riley delivered the baby girl in an emergency caesarean section, and Ms. Brown was taken to intensive care. Doctors at River Park, the only hospital in a central Tennessee county of 40,000 people, didn’t know what had caused Ms. Brown’s seizure. But they knew one thing: The 28-year-old woman needed more than they could provide.

What followed was a race to save Ms. Brown, a high-risk medical challenge that would involve frantic requests for transportation, an hour-and-a-half ambulance ride through mountains and the rain, and last-minute medical interventions as she tore through the hospital’s blood supplies.

Since the start of the century, it has become more dangerous to have a baby in rural America. Pregnancy-related complications are rising across the U.S., and many require specialized care. For some women, the time and distance from hospitals with the resources and specialists to handle an obstetric emergency can be fatal.