A boat overloaded with Cubans being smuggled into the U.S. tried to ram a Coast Guard vessel in rough seas early Saturday, and a woman aboard the boat died, authorities said.

The 36-foot, go-fast boat ignored orders to stop when the Coast Guard tried to intercept it 4 miles south of Boca Chica around 6:30 a.m., said Petty Officer James Judge, a spokesman for the Coast Guard.

Coast Guard crew then fired two shots into the vessel's engine to disable it, he said.

"The boat was bouncing around like crazy. It was very rough, choppy waters," Judge said. "They repeatedly attempted to ram the vessel more than five times, but they never made contact."

The boat carried 31 Cubans and three people authorities said were smugglers.

The woman who died suffered a head injury and severe bruising to her face, and an autopsy was scheduled Sunday.

Judge said three men on the boat were treated for minor injuries. Authorities said none of the injuries was caused by the Coast Guard gunfire. A pregnant woman also was hospitalized, but her condition was not immediately known.

The remaining migrants were still aboard the Coast Guard cutter and will be processed as usual, he said. Under the "wet foot/dry foot" policy, most Cubans who reach U.S. soil are allowed to remain, while those intercepted at sea are generally sent home.

It was not immediately known where in Cuba the migrants began their journey. Go-fast boats from Cuba can typically reach South Florida in a couple hours, Judge said.

"The boat was severely overloaded," he said. "All those add up for dangerous possibilities."

Authorities detained three people as suspected smugglers, though criminal charges were not immediately filed.

"Smugglers often treat migrants as if they were human cargo, with blatant disregard for individual life and safety. This must stop," U.S. Attorney R. Alexander Acosta said in a statement. "The United States Attorney's Office is determined to use all prosecutorial tools at our disposal to investigate and prosecute these smuggling incidents."

A spokesman for the Cuban American National Foundation said the shots fired by the Coast Guard were "of grave concern."

"The people who are culpable are those who engage in the smuggling of humans," said Alfredo Mesa, the group's executive director. "At the same time, we call upon the U.S. Coast Guard to remember that these are human beings fleeing tyranny."