The Liverpool-born comedian, 54, said he was driving through Wales on his way to catch a ferry to Ireland when the bizzare road accident happened.
Bishop shared photos on Instagram of his Land Rover smashed into a metal post at the side of the road and the windscreen shattered.
He confirmed that he missed the ferry anyway and instead travelled as a foot passenger on a later crossing.
He wrote: “I was driving to catch a ferry to Ireland when something happened!!
“To avoid a car that was swerving to avoid a ‘big chicken’ as I overtook I ended up like this.
“Nobody was hurt thanks to @landrover_uk being brilliant and no oncoming traffic.
“I missed my ferry but went on as a foot passenger on a later one thank to James and Tony from @dyfedpowyspolice .
“Amazing how helpful people can be when you need them and thanks to the other drivers for staying including Andrew who I think sadly hit the ‘big chicken’ anyway.”
He will appear in the new series of the BBC One show, in which Jodie Whittaker reprises her role as the Time Lord.
He plays a character called Dan, who “becomes embroiled in the Doctor’s adventures”.
Bishop will star alongside Whittaker and Mandip Gill, who returns as Yaz.