The car did not stop and continued into Shawstead Road, where the driver was arrested having left the vehicle on foot.
A quantity of cannabis was later recovered by police from a roundabout on North Dane Way.
Investigating officer, PC Hannah Cooper, said: ‘We are appealing for witnesses who may have seen the VW Golf travelling along North Dane Way and a package being thrown from the window.
‘In particular, drivers with dashcam are asked to check for footage that may assist the investigation.’
A 25-year-old man from Chatham was later released pending further investigation into allegations of drug supply, failing to stop for a police officer and driving offences.
Anyone with information is urged to call Kent Police on 01634 792209, quoting reference 46/74026/22.
You can also call Crimestoppers anonymously on 0800 555111 or complete the online form on their website.