The claim came as the association, a trade body representing around 400 UK design firms, released a survey into its members' concerns about immigration rules after Brexit.
It called for a rethink of immigration rules, which one member described as "onerous, time-consuming and costly".
DBA chief executive Deborah Dawton said she had heard of UK designers refusing to return home due to fears over even tougher immigration rules in future.
"A lot of UK designers will make it into senior roles in global businesses," she told Dezeen. "The challenge then is that if they want to move back to the UK but they've got non UK national families or partners, then it's quite possible that's going to throw a spanner in the works for them."
The survey, carried out in response to a government call for industry input on post-Brexit immigration rules, found that EU workers make up 15 per cent of DBA members' workforce, with half of the firms surveyed employing at least one person from the continental EU.
Hiring overseas talent allows firms to compete globally
The key reason for hiring overseas talent is to allow firms to compete globally, the survey said.
"Their priority is the ability to continue to compete effectively in the global marketplace, which requires access to a culturally diverse workforce of world-class talent, at all levels of experience," it said.
"It is not only technical skills that matter," said a large Scottish design agency in its response to the survey.
"Cultural and creative diversity is essential to our industry. If we only had access to UK nationals it would limit our global understanding and our ability to export and trade internationally. It would mean we would probably need to set up offices abroad employing locally rather than generating jobs within the UK."
Firms said they are increasingly getting work from overseas clients, meaning they needed staff with language skills and local cultural knowledge. Many pointed out that they were not hiring foreign workers because they were cheaper but because they needed a diverse and experienced workforce.
"With an increasing number of clients based from outside of the UK, language skills are very important to the company in order to secure projects with non-English speaking clients," responded a large retail design agency based in London.
"In an increasingly competitive market we are looking to attract candidates with the best skills and experience and need to be able to recruit from a wide pool of candidates, which includes EU nationals."
However DBA members reported that existing immigration rules are "very complex for small businesses" and that some firms avoid recruiting talent from outside the EU due to the cost and delay of the application process. After Brexit, EU workers may be subject to similarly tough immigration rules.
"The current visa system is onerous, time-consuming and costly," said a digital agency based in Yorkshire. "The digital infrastructure behind the application and renewal system is clunky and the human interaction with the Home Office is pretty unhelpful."
"UK design has earned a formidable global reputation, built on the depth and quality of talent and expertise found across the sector, which – as the DBA survey for the DCMS clearly shows – benefits from being a culturally diverse workforce," Dawton said.
"An immigration system that considers the country’s business needs, and encourages future growth and competitiveness is imperative."
Businesses are already dealing with immigration issues posed by Brexit
The report found that half of the DBA's member businesses are already dealing with immigration issues posed by Brexit.
"There's a perception that people aren't welcome in the UK anymore," said Dawton.
"It's one thing to be open for business and it's quite another to be welcoming to business. And if we start to make it difficult for people, they will focus their efforts on countries that are more welcoming."
According to Dawton, the prospect of Brexit is affecting the UK's ability to hold onto talent, both foreign and home-grown.
One senior British designer, who is head of innovation for a US company, recently turned down a job in the UK because his wife is German, and he was worried that the move would cause problems for her.
"He wasn't willing to play Russian roulette with his family," she said.
Similarly, a small design agency in the Midlands revealed that it was worried about losing a talented Polish employee, who felt "less welcome, less secure and less loyal to a country with a national press that seems to delight in belligerent hostility towards anyone and anything 'foreign'."
"If you couple this with the changes we've seen in the education system, we're likely to end up with a hole in the market," said Dawton, referring to the falling number of UK students taking creative courses at school.
Firms call for a boost in government investment in arts courses
More than half of respondents said the government would need to boost creative education by increasing investment in arts courses, expanding the commercial focus of degree courses and improving the teaching of digital skills.
However the number of UK students taking creative subjects has suffered an "alarming" fall, which many blame on the introduction of the English Baccalaureate curriculum, which emphasises academic subjects. This has led to fears of an impending skills crisis in the creative sector.
"Developing home-grown talent takes years, and we need to plug the gap of those people who are no longer applying for jobs in the UK."
Released this week, the DBA Immigration Survey was commissioned by the government's Department for Digital, Culture, Media and Sport (DCMS).
In the survey, DBA members asked the government to maintain freedom of movement for EU nationals or, failing that, to preserve existing rights for EU employees.
They also asked for a "straightforward and inexpensive processes that allow businesses to employ whom they believe is best for the role in their business" and called on ministers "to avoid short-sighted [immigration] quotas aimed at hitting politically motivated targets and to instead consider the country’s business needs".
Almost 90 per cent of EU workers in the design sector are on permanent contracts, the survey found, with almost three quarters working at mid- or senior-level within firms.
Almost all – 99 per cent – were employed because they were "the best applicant for the job", rather than because they had a specific skill set which could not be found in the UK.
"I don't think there is anything particularly surprising about the results that came back," said Dawton.
"If anything, it helps to emphasis the fact that one of the key aspects of the creative industries in the UK is its culturally diverse workforce," she added. "Our concern is that this cultural diversity [will be] impeded going forward and that businesses that are currently recruiting from outside the EU are going to start finding it increasingly difficult to do that."
Design represents 7.2 per cent to the UK economy and employs more than 1.5 million people, according to the DBA, while an independent review of the creative sector published last week said a further one million jobs could be created by 2030.
However Peter Bazalgette, the author of the review, argued that "you don't need to have a creative industries degree to work in the creative industries," saying young people were already exposed to creative processes thanks to mobile phones.