The Metropolitan Police Service said on Twitter that the man had been pronounced dead after being shot at around 2 pm (1400 GMT) in the Streatham neighbourhood.
“We believe there are two injured victims.
“We await updates on their conditions,” the police wrote.
The scene has been “fully contained,” police said, but they asked the public to avoid the area while the investigation continued. The London Ambulance Service said it had “a number of resources” at the scene.
“The circumstances are being assessed; the incident has been declared as terrorist-related,” police said.
A witness to the attack told Britain’s Press Association that the man was carrying a machete and had “silver canisters on his chest.” The witness said the suspect was shot near a Boots cosmetics and health care retailer on Streatham High Road.
The attack was claimed by the Islamic State extremist group.
That attack echoed an earlier one in June 2017, when three men drove a van into pedestrians on the bridge before stabbing people at bars and restaurants in the nearby Borough Market.
Eight people died and 48 were injured in the rampage, which was also claimed by the Islamic State.
In August 2018, a man swerved his car into cyclists, pedestrians and police officers in front of the Houses of Parliament, injuring several people.