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The number of cars made in the UK during March fell by 13.3% as both domestic and overseas demand for vehicles declined.
The Society of Motor Manufacturing and Traders (SMMT) said poor weather had affected production.
But it said car sales to the UK had dropped for an eighth month and double-digit falls in the home market and exports were a "considerable concern".
It said the UK must stay in the customs union to ensure "frictionless trade".
The SMMT said that cars made for the UK fell by 17.7%, while exports dropped by 11.9%.
Some carmakers have announced plans to cut jobs and reduce production amid uncertainty over the UK's exit from the European Union, as well as higher taxes on diesel vehicles.
Earlier this month, Jaguar Land Rover said it would not renew the contracts of 1,000 temporary workers at two of its factories because of "continuing headwinds" affecting the car industry.
Meanwhile, Nissan said it would cut hundreds of jobs at its plant in Sunderland.
Over the first three months of the year, car production fell by 6.3% to 440,426.
Exports fell by 4% in the first quarter, while UK demand tumbled by 14.1%.
Mike Hawes, chief executive of the SMMT, said: "Following recent announcements on jobs cutbacks in the sector, it's vitally important that the industry and consumers receive greater certainty, both about future policies towards diesel and other low-emission technologies, and our post-Brexit trading relationships and customs arrangements.
"Maintaining free and frictionless trade is an absolute priority - it has been fundamental to our past success and is key to our future growth."