The 23-year-old sustained the injury while attempting a tackle during training with the Springboks in London.
The 23-year-old sustained the injury while attempting a tackle during training with the Springboks in London and is awaiting the results of a scan, but looks set to be ruled out of South Africa's Tests against England, Italy and Wales.
It is another blow for South Africa after Jesse Kriel was ruled out of the three Tests with a leg injury sustained during Saturday's 31-31 Wembley draw with the Barbarians.
Kriel has been replaced by Rohan Janse van Rensburg for the remainder of the tour and coach Allister Coetzee must now find another replacement for Smit.
"He's at the hospital and we are awaiting the result of a scan but it looks like a possibility he might lose out on the rest of the tour," Coetzee said.
"It could be a bicep tear. We'll get confirmation but it might be the end of his tour. He was attempting a tackle and there was a hyperextension of the arm."
He added: "We have an openside crisis in the country.
"Francois Louw, Jaco Kriel, Heinrich Brussow and others aren't available, but I'm comfortable we have cover."
Coetzee is under pressure following South Africa's recent poor run of results, including a record 57-15 defeat by New Zealand last month.
They needed two late tries to force a draw with the Barbarians and will be underdogs when they take on England at Twickenham this weekend.
But South Africa have not lost to England since 2006 and Coetzee is boosted by the arrival of several of his more experienced players, including JP Pietersen, Bryan Habana and Warren Whiteley, who all joined up with the squad on Sunday.
"A week is a long time in rugby. We feel like we started last night, we had a full complement of the Springboks team at our first team meeting," Coetzee said.
"You get the feeling it's a big week, the start of the so-called official tour. We are raring to go.
"We are definitely the underdogs, we are playing the second-best team in the world who are on a nine-game winning streak.
"They won all three Tests in Australia so there is a lot of confidence in the English side."