Police spotted a silver Subaru Outback driving the wrong way on the Whitchurch slip road northbound on the A34.
Officers attempted to stop the vehicle, which then joined the A34 from the slip road and began driving in the correct direction.
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A police spokeswoman said: ‘Following a pursuit officers lost sight of the vehicle at the Beacon Hill exit.
‘We have been conducting enquiries to locate the vehicle involved and are now turning to the public for help.
‘Were you in the area at the time? Did you see a silver Subaru Outback driving the wrong way up the slip road? Did you witness the vehicle travelling along the A34 afterwards?’
Officers want to hear from anyone with dash cam footage of the incident or the vehicle around the time it happened – which was 3.20pm on Tuesday.
They think the car may have damage to its offside and no front number plate.
Anyone with information is asked to contact police on 101 quoting reference 44210430064.
Alternatively, submit information through this link.