Rachel Rojas, the special agent in charge of the FBI's office in Jacksonville, said the FBI was working, as it does in most mass shootings, on the presumption that it was an act of terrorism, but she stressed that was largely to allow investigators to use special tools afforded to them in terrorism cases.
"We are looking very hard at uncovering his motive and I would ask for patience so we can get this right," she said.
The FBI identified the shooter as Second Lieutenant Mohammed Saeed Alshamrani, 21, and said he opened fire inside a classroom at the base early on Friday morning.
He killed three people and wounded eight before being fatally shot by a deputy sheriff, the FBI said on Sunday.
Investigators are also trying to establish whether he acted alone or was part of a larger plot.
His fellow Saudi students were speaking directly with American investigators and were restricted to the base on order of the Saudi military, Rojas said.
"I thank the kingdom for their pledge of full and complete cooperation," she added.
American authorities said no Saudis have so far been arrested in the case.
US President Donald Trump said on Saturday that King Salman and Crown Prince Mohammed bin Salman were "devastated" by what took place and pledged to help families of the victims.
The Saudi crown prince called Trump on Sunday to assure him Saudi authorities would offer their absolute cooperation with the United States and provide all information that would help the investigations, the Saudi state news agency reported.
Alshamrani was on the base as part of a US Navy training program designed to foster links with foreign allies.
He had started training in the United States in 2017 and had been in the Pensacola area for the past 18 months, authorities said.
The US has long had a robust training program for Saudis, providing assistance in the US and in the Kingdom. More than 850 Saudis are in the United States for various training activities. They are among more than 5,000 foreign students from 153 countries in the US going through military training.
