The blaze broke out in The Downs just after 11 o’clock.
Most of a fourth-floor flat in the four-storey block was damaged.
There have been no reports of any injuries.
London Fire Brigade was called at 11.23 and the fire was brought under control by 12.17.
Crews from Wimbledon, New Malden, Wandsworth, Mitcham and surrounding fire stations have been in attendance.