No injuries were reported.
A Wiltshire Police spokesman said: “At around 4.20am today (May 18) we received a call regarding a single-vehicle road traffic collision on the High Street in Malmesbury.
“It involved a black Renault and there was some damage to the bridge, which caused the road to be closed.
“The driver, a 41-year-old woman from Bristol, was arrested on suspicion of drug driving and is currently in custody at Melksham Police Station.
“Nobody else was in the vehicle.”
Wiltshire Council has been contacted regarding the road closure.