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The incident occurred at a nightclub in the resort. The soldier, a 19-year-old serving with the Second Battalion, Royal Regiment of Fusiliers, was stabbed in the chest and was dead upon arrival at a local hospital.
Police identified the victim as David Lee Collins, 19.
Two police sources told Reuters that the three suspects held for questioning were British citizens, all teenagers. They were due to appear before a magistrate on Monday, police said.
"We can confirm there was a stabbing incident involving a British soldier who was killed," a spokeswoman for the British bases in Cyprus said.
The victim was serving at Dhekelia, one of two bases Britain has maintained on the east Mediterranean island, a former colony, since independence in 1960. The incident happened in an area of Ayia Napa which is out-of-bounds for British soldiers.