Georgia head coach Levan Maisashvili is in a Johannesburg hospital with "serious lung damage" after testing positive for Covid-19.
A Georgia Rugby statement said the head coach "is on artificial respiration".
"Everything is being done to improve Levan Maisashvili's condition," it added.
The union said Maisashvili had "been moved to one of South Africa's top-level clinics".
"The rest of the team has already returned to Georgia while Levan and the team doctor will stay in South Africa for as long as needed," the statement continued.
"Five more Covid-positive team members are finishing their recovery and will leave for Georgia on Sunday."
The Springboks played Georgia in preparation for the British and Irish Lions Test series, which begins on 24 July.