Theresa May gave them the guarantee, amid continuing sovereignty claims by Argentina, 35 years after its failed invasion of the British overseas territory.
It came in her Christmas radio message to the remote South Atlantic archipelago. The Falklands War in 1982 ended with the islands returned to British control after an Argentine surrender.
During the fighting 255 British and about 650 Argentine servicemen were killed, plus three Falkland islanders.
In 2013, residents voted overwhelmingly for the island to remain a UK possession.
In her personal seasonal message, the Prime Minister said: “This year marked the 35th anniversary of the conflict to preserve your freedom and independence.
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“I want you to know that I will never allow anyone to compromise your right to self-determination – a right you freely expressed in the 2013 referendum; and a legacy for which so many paid the ultimate price.”
However, she said that there were “non-sovereignty related areas” where the UK would work with Argentina for mutual benefit, including a commitment to lift “restrictive measures” imposed on the islands.
Mrs May said her Christmas message was very simple.
“You can count on me as a UK Prime Minister who will do everything possible to support you in realising the full potential of these islands.”