Arla Foods, which has already warned of price consequences for UK consumers in the event of a hard Brexit, said a softer version could also leave costs soaring because of the risk of shortages.
The company, which counts Anchor butter among its brands, spoke up as it published research it had commissioned from the London School of Economics.
The Danish-based co-operative said it found that the potential for customs-related delays risked raising the cost of everyday dairy goods such as butter and yoghurt and cheese, with more specialised products becoming luxury items.
It identified several other price pressures including knock-on effects across the supply chain for products with limited shelf-life and the possibility Brexit resulted in restricted access to vets, lorry drivers and farm workers.
Another was the growth in red tape for exporters and importers.
The LSE report estimated that even a seven-minute additional waiting period for each customs inspection would add £111 to the cost of a container of goods because of extra labour costs.
Arla spoke out as UK food manufacturers – especially in the soft fruit sector – warn that the potential for a lack of access to labour from eastern Europe threatens domestic production and the ability of farmers to help fill any import voids.
The company said the UK currently has the second-largest dairy trade deficit with the rest of the world – with 98% of dairy imports currently originating in the EU.
There have been similar warnings over a wider UK reliance on food imports.
Arla’s UK managing director, Ash Amirahmadi, said: “The farmers that own the Arla dairy cooperative already balance keeping consumer prices down with maintaining quality and the best standards, including high animal welfare.
“There’s no margin to play with here in the value chain. Any disruption means that if we don’t get the practicalities of Brexit right we will face a choice between shortages, extra costs that will inevitably have to be passed on to the consumer or undermining the world-class standards we have worked so hard to achieve.”
Sky News has approached the government for a response.