The shootings took place in the northern Californian city of Oakland after around 7,500 protesters took to the streets in protest at the death of Mr Floyd at the hands of a police officer.
“Two Federal Protective Services officers stationed at the Oakland Down Town Federal Building suffered gunshot wounds. Unfortunately, one succumbed to his injury,” the Oakland police department told CNN.
No suspects have been arrested and an investigation has been launched.
As well as Minneapolis and Oakland, protests have broken out in New York, Washington D.C., Atlanta, Detroit, Portland and many other cities.
Confrontations have often been violent. A teenager was killed in Detroit after shots were fired on a crowd of people on Friday.
The US army has asked military police units from New York and North Carolina to be on standby to intervene in the protests.
In Washington D.C. protesters reportedly stayed out all night outside the White House, while the Secret Service - the President's bodyguards - stood guard.
Donald Trump said that if protesters had got through the guards, they would have been attacked by "vicious dogs" and "ominous weapons".
He tweeted: "They were not only totally professional, but very cool. I was inside, watched every move, and couldn’t have felt more safe.
"They let the 'protesters' scream & rant as much as they wanted, but whenever someone got too frisky or out of line, they would quickly come down on them, hard – didn’t know what hit them."
Great job last night at the White House by the U.S. @SecretService. They were not only totally professional, but very cool. I was inside, watched every move, and couldn’t have felt more safe. They let the “protesters” scream & rant as much as they wanted, but whenever someone....— Donald J. Trump (@realDonaldTrump)May 30, 2020
He added: "The front line was replaced with fresh agents, like magic. Big crowd, professionally organized, but nobody came close to breaching the fence. If they had they would have been greeted with the most vicious dogs, and most ominous weapons, I have ever seen."
Derek Chauvin has been charged with third-degree murder and manslaughter and fired from the police after he was filmed kneeling on Mr Floyd's neck for at least eight minutes while arresting him for allegedly using a counterfeit bank note.