The driver of the quadbike was caught on Christchurch Road, New Milton.
The vehicle was not insured or road legal, said a spokesperson for Hampshire Constabulary.
It comes as the force told followers on social media of the “multitude of incidents” officers had been dealing with throughout the national park during the week.
Police arrested a man for being drunk and disorderly after he was reported to have been “actively abusing members of the public.”
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Officers also found a stolen van in Poulner, Ringwood, before returning the vehicle to its owner, as well as dealing with various shoplifting offences.
In a statement, a spokesperson for Hampshire police said: “We have been busy dealing with a multitude of incidents as always.
“These included seizing a quadbike that was being ridden down Christchurch Road at speed and wasn’t insured or road legal. It was reported to be causing anti-social behaviour in the area.
“Officers also arrested a male for being drunk and disorderly after he was reported to have been actively abusing members of the public.
“As always, these incidents are just a few from many.”