The incident happened in Rye Lane near Peckham High Street at around 11.20am yesterday (December 20).
Officers and London Ambulance Service crews attended and found a woman with an injury to her head.
She was taken to hospital where her injuries are believed to be non-life-threatening but potentially life changing.
The rider of the e-bike did not stop at the scene following the incident.
No arrests have been made as police enquiries continue.
A spokesperson for the Metropolitan Police said: “Anyone with information that could assist police is asked to call 101 quoting CAD4585/20DEC.
“Alternatively, call the independent charity Crimestoppers anonymously on 0800 555 111.”
