The star, 54, said he was driving through Wales to board a ferry to Ireland when the crash happened.
The Doctor Who actor shared photos on social media of a car - a Land Rover - smashed into a post at the side of the road. It had a broken windscreen.
Thankfully, no one was hurt in the crash.
The star said he missed the ferry due to the crash and had to tracel as a foot passenger on a later crossing.
He wrote on his Instagram page: “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 Land Rover being brilliant and no oncoming traffic.
“I missed my ferry but went on as a foot passenger on a later one thanks to James and Tony from Dyfed Powys Police.
“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.”
It was confirmed that Bishop was the latest addition to the Doctor Who cast earlier this year.
Bishop will appear in the popular fantasy BBC One show alongside Jodie Whittaker, who reprises her role as the Time Lord.
He plays a character called Dan, who “becomes embroiled in the Doctor’s adventures”.
Mandip Gill, will also return as character Yaz.