UPDATE 2-Japan warns on Brexit: we cannot continue in UK without profit

* UK PM met major Japanese tech firms, banks, carmakers

* Ambassador warns on need to maintain free trade

* PM, ministers said nothing firm on free post-Brexit trade-source (Recasts with ambassador, adds details)

LONDON, Feb 8 (Reuters) - Japan warned Prime Minister Theresa May on Thursday that its companies would have to leave Britain if trade barriers after Brexit made them unprofitable.

Japanese firms have spent more than 40 billion pounds ($56 billion) in Britain, encouraged by successive governments since Margaret Thatcher which promised a business-friendly base from which to trade across the continent.

After May and several of her top ministers met bosses from 19 Japanese businesses, including Nissan, SoftBank and bank Nomura, Japan's ambassador to Britain issued an unusually blunt warning on the risks of trade barriers.

"If there is no profitability of continuing operations in the UK - not Japanese only - then no private company can continue operations," Koji Tsuruoka told reporters on Downing Street when asked how real the threat was to Japanese companies of Britain not securing frictionless trade.

"So it is as simple as that," he said. "This is all high stakes that all of us, I think, need to keep in mind."

Japan, the world's third largest economy, has expressed unusually strong public concerns about the impact of Brexit on the United Kingdom, the second-most important destination for Japanese investment after the United States.

In a warning after the shock 2016 Brexit vote, Japan expressed fears about a cliff edge that could disrupt trade when Britain formally leaves the bloc in March 2019.

Major corporations have sought a two-year transition period, which they hope will bridge Britain into its new relationship with the bloc and allow businesses to make preparations for any trade or regulatory changes.

Both London and Brussels hope to agree a transition deal lasting until the end of 2020, in which Britain would remain in the single market and be bound by all EU laws, by a March 22-23 summit.

May and her ministers assured Japanese businesses of the importance of maintaining free and frictionless trade after Brexit during the meeting but said nothing firm on the matter, a source familiar with the discussions told Reuters.

"The point about frictionless trade and tariff-free trade was made in the meeting and acknowledged by the government and all sides as being important but nothing firm," said the source, who spoke on condition of anonymity.

The meeting with Japanese investors came after ministers discussed their Brexit strategy including how closely Britain should remain aligned with the EU and its customs union, a divisive issue for the ruling Conservatives.

Hitachi Europe's Deputy Chairman Stephen Gomersall, Mitsubishi CEO for Europe and Africa Haruki Hayashi, SoftBank Investment Advisers UK CEO Rajeev Misra and Nomura's Executive Chairman in Europe, the Middle East and Africa Yasuo Kashiwagi joined the meeting.

Nissan's Europe Chairman Paul Willcox, Honda's Senior Vice President in Europe Ian Howells and Toyota's Europe President and Chief Executive Johan van Zyl were also present.

Collectively the three carmakers build nearly half of Britain's 1.67 million cars and the industry has warned that any loss of free and unfettered trade could hit its operations.

Nissan announced in 2016 that it would build two new models at its site in Sunderland, northern England, after what a source said was a government promise of extra support in the event that Brexit hits the competitiveness of the plant.

(Additional reporting by Elizabeth Piper; editing by Guy Faulconbridge)