Mandla Maseko, 30, was killed on Saturday, a family statement says.
In 2013, the South African Air Force member beat one million entrants to win one of 23 places at a space academy in the US.
Nicknamed Afronaut and Spaceboy, Maseko described himself as a typical township boy from Pretoria.
He had spent a week at the Kennedy Space Centre in Florida doing tests in preparation for an hour-long sub-orbital flight, originally scheduled for 2015.
Maseko said he wanted to do something that would motivate and inspire young people in Africa and prove that they could achieve anything whatever their background.
He told the BBC he planned to call them from space. "I hope I have one line that will be used in years to come - like Neil Armstrong did," he said.
The US astronaut, who died in 2012 aged 82, was the first man ever to walk on the Moon in 1969.
As he stepped on to the lunar surface, he famously said: "That's one small step for man, one giant leap for mankind."