Moon Climbing, which makes clothes and training equipment, shifted some operations abroad after ‘multiple difficulties’ and confusion over documentation, VAT and red tape.
Boss Ben Moon said he expected it to become their main base to serve the EU and the rest of the world.
He added: “What is absolutely clear is that we do not have ‘frictionless trade’.”
An EU/UK free trade agreement came into force in January after the UK left the European Union.
Sheffield Central MP Paul Blomfield said several other businesses had contacted him to say they’d had to either move abroad or change operations or supply chains because of new Brexit barriers.
And he urged the Government to do more to protect local businesses.
He told the House of Commons: “Moon Climbing, a specialist rock climbing supplier in my constituency, tells me that since January, new barriers have damaged their trade with Europe.
“In line with the apparent advice of DIT officials, they’ve set up a base in the Netherlands to avoid the barriers – and they anticipate it ‘will be our main base from which we service both the EU and the rest of the world’.
“I heard the Minister and the Secretary of State say earlier that it’s nothing to do with them, but frankly companies expect the Department for International Trade to take some responsibility for trade.
“So what are they doing to prevent more UK businesses moving abroad as a result of the damaging Brexit deal - losing UK jobs, GDP and tax revenue?”
Graham Stuart, Under Secretary of State at the Department of International Trade, said frictions were ‘inevitable’ but the government had given people the tools to overcome them.
“The British people decided to leave the European Union and we are supporting businesses in Europe and beyond.
“It is not overly complicated to accept it is the Cabinet Office and the unit led by Lord David Frost who are taking responsibility for those negotiations.
“But we work actively and we run webinars with thousands attending in order to give people the tools to overcome the frictions which inevitably result from our departure.