Between 5am and 5.30am on Sunday (November 24), a grey Audi was reported to have been driving at high speed with no lights on the coastbound carriage on the motorway.
The driver had been driving between Junction 5 at Aylesford and Junction 10 at Ashford.
A 42-year-old man was arrested in relation to the incident and has since been charged with offences including dangerous driving and drink driving.
Officers believe the drivers of other vehicles may have seen and potentially recorded the incident on dashcams and are urging anyone who has yet to speak to them to call Kent Police on 01843 222289, quoting reference 46/200527/24.
Drivers can also call Crimestoppers anonymously on 0800 555111 or complete the online form.
