London Fire Brigade (LFB) has sent 25 engines to the tackle the fire on Harry Zeital Way, Upper Clapton.
Part of an balcony is alight from the ground to the fifth floor, the brigade said.
Dramatic video posted on Twitter showed flames rising up the side of the building.
LFB was first alerted to the fire at 10.39pm. It said it had taken 67 calls about the blaze.
Crews from Stoke Newington, Homerton, Bethnal Green, Islington, Holloway, Shoreditch and other surrounding fire stations were sent to the scene.
The cause of the fire is not yet known.
One woman was led to safety by firefighters but there are no reports of injuries , LFB said. Crews are expected to be at the scene throughout the night.
Additional reporting by agencies
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