Iran’s Islamic Revolution Guards Corps (IRGC) said the aircraft had violated Iranian airspace, and that the incident sent a “clear message to America”.
But the US military insisted the drone had been over international waters at the time, and condemned what it called an “unprovoked attack” by the IRGC.
President Donald Trump tweeted: “Iran made a very big mistake!”
The incident comes at a time of escalating tension between the two countries.
On Monday, the US defence department said it was deploying 1,000 extra troops to the region in response to “hostile behaviour” by Iranian forces. It has already sent an aircraft carrier strike group and B-52 bombers.
The US has also accused Iran of attacking two oil tankers with mines last Thursday just outside the Strait of Hormuz, in the Gulf of Oman. Iran rejects the allegation.
It was the second time in a month tankers had been attacked close in the region, through which a fifth of the world’s oil passes each day.
Tensions were further fuelled on Monday when Iran announced its stockpile of low-enriched uranium would next week exceed limits it agreed with world powers under a landmark nuclear deal in 2015.
Iran stepped up its production in response to tightening economic sanctions from the US, which unilaterally withdrew from the deal last year.
What happened on Thursday?
The IRGC said its air force had shot down a US “spy” drone in the early hoursafter the unmanned aircraft violated Iranian airspace near Kuhmobarak in the southern province of Hormozgan.
In a speech carried live on Iranian state television, IRGC commander-in-chief Maj-Gen Hossein Salami warned the US to respect Iran’s territorial integrity.
“Those who defend the borders of the Islamic nation of Iran will react in a total and decisive way to any intrusions by foreign elements on our land. Our borders are our red line.”
He added: “Iran is not seeking war with any country, but we are fully prepared to defend Iran.”
The US military’s Central Command confirmed a US Navy Broad Area Maritime Surveillance (BAMS-D) aircraft was shot down by an Iranian surface-to-air missile system while operating in what it said was international airspace over the Strait of Hormuz at 23:35 GMT on Wednesday (04:05 Iran time on Thursday).
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“Iranian reports that the aircraft was over Iran are false,” spokesman Navy Capt Bill Urban said. “This was an unprovoked attack on a US surveillance asset in international airspace.”
Reuters news agency later cited a US source as saying US naval assets had been dispatched to the drone debris field in international waters.
The BAMS-D is a RQ-4A Global Hawk High-Altitude, Long, Endurance (HALE) drone that can carry out surveillance and reconnaissance missions over vast ocean and coastal regions, according to the US military.