Four people are dead, including the gunman, after a shooting Friday morning at Naval Air Station Pensacola, authorities said. Several other people were injured, including two sheriff's deputies who engaged the shooter.

The suspect was a member of the Saudi Air Force and an aviation student at the base, officials confirmed at a press conference Friday. CBS News has learned the suspect's name is Mohammed Saeed Alshamrani, although the FBI said they would not confirm a name during a Friday night press conference.

The Pentagon told CBS News that the shooter had been in the country since August 2017 pursuant to a U.S. Air Force Foreign Military Sales training case funded by the Kingdom of Saudi Arabia and that he was scheduled to stay until August 2020. He was enrolled in English, basic aviation, and initial pilot training at the station, the Pentagon added.

Get Breaking News Delivered to Your Inbox

Investigators are trying to determine how a foreign national obtained a handgun, which is prohibited by law, and carried it onto the base. FBI special agent in charge Rachel Rojas said officials have not yet determined a motive for the shooting.

U.S. Attorney for the Northern District of Florida Larry Keefe said officials were working tirelessly to uncover more information. "We will be absolutely relentless," he said, adding "We will not stop, we will work 24/7, there is nothing more important than this to us."

Federal law enforcement sources confirmed to CBS News' Jeff Pegues that investigators have talked to other Saudi nationals in connection with the shooting. The sources said no one has been detained, and no one else is being called a suspect at this time.

According to the Middle East Media Research Institute, someone matching the shooter's identity had an online footprint, including a Twitter post that criticized America's support for Israel.

"O American people, I'm not against you for being American, I don't hate you because your freedoms, I hate you because every day you supporting, funding and committing crimes not only against Muslims but also humanity," the post read.

"What I see from America is the supporting of Israel which is invasion of Muslim countrie (sic), I see invasion of many countries by it's troops, I see Guantanamo Bay. I see cruise missiles, cluster bombs and UAV."

Base commander Captain Tim Kinsella explained that the base has a longstanding program to train military students from ally nations. Military from around the world go to the base for flight training.

"There's always been international students training here," Kinsella told reporters. "He was in the aviation pipeline ... training in aviation."

The Pentagon said that there are currently 5,181 foreign students from 153 countries in the United States for Department of Defense security cooperation related training.

President Trump said Friday that Saudi King Salman called to express his condolences and sympathies.

"The King said that the Saudi people are greatly angered by the barbaric actions of the shooter, and that this person in no way shape or form represents the feelings of the Saudi people who love the American people," he tweeted.

King Salman of Saudi Arabia just called to express his sincere condolences and give his sympathies to the families and friends of the warriors who were killed and wounded in the attack that took place in Pensacola, Florida.... — Donald J. Trump (@realDonaldTrump) December 6, 2019

The Saudi state news agency said King Salman affirmed when speaking to President Trump that the perpetrator of the "heinous crime" does not represent the Saudi people. King Salman instructed Saudi security services to cooperate with U.S. agencies in the investigation, the state news agency reported.

"I think there's obviously going to be a lot of questions about this individual being a foreign national, being a part of the Saudi Air Force - to be here training on our soil - to do this," Florida Governor Ron DeSantis said at the press conference. "The FBI is working with DOD ... to answer those questions."

"One of the things that I talked to the president about is ... the government of Saudi Arabia needs to make things better for these victims," DeSantis said. "And I think that they're going to owe a debt here given that this was one of their individuals."

Escambia County Sheriff David Morgan said at a press conference earlier Friday that "the threat has been negated." He said authorities were not looking for any additional shooters.

"While we would love to do a full disclosure of what occurred this morning about everything that we are aware of, quite obviously we cannot do that. There are things – we are now in the investigative process – that are required to be held back," Morgan said.

"Walking through the crime scene was like being on the set of a movie," he said.

"You just don't expect this to happen at home. This doesn't happen in Escambia County, this doesn't happen in Pensacola, this doesn't happen to our friends and neighbors in the United States Navy. But it did."

Emergency responders near the Naval Air Base Station in Pensacola, Florida, after a shooting at the base on December 6, 2019. WEAR-TV via AP

The shooting took place in Building 633, which is part of the Naval Aviation Schools Command and is where Navy personnel entering flight training first report. Two Escambia County deputies shot and killed the gunman, and by 7:50 a.m., the threat was over. Both deputies were wounded — one in the knee, another in the arm.

Officials confirmed the shooting was contained within two floors in one building.

The base said it will be closed for the remainder of Friday and that only "essential personnel" will be allowed to enter.

The FBI is taking over the investigation.

Naval Air Station Pensacola is a United States Navy base located next to Warrington, Florida. The base employs more than 16,000 military and 7,400 civilian personnel. It is also home to the Blue Angels flight demonstration team and the National Naval Aviation Museum.

This is the second shooting to occur on a U.S. Navy base this week. On Wednesday, a sailor opened fire at a U.S. Navy shipyard at Joint Base Pearl Harbor-Hickam in Hawaii, killing two civilian shipyard employees and wounding another before dying from an apparent self-inflicted gunshot wound, according to a Navy spokesman.

Jeff Pegues, Mark Strassmann and David Martin contributed to this report.