Tehran, Iran - Iran's supreme leader vowed to teach the US "unforgettable lessons" on Saturday, as Washington reported its first military deaths since renewing its hostilities with the Islamic republic.

A month after the foes signed a now-abandoned preliminary deal aimed at ending their war, Tehran struck infrastructure around the Gulf in retaliation for a week of intensifying US attacks, which Iran said had hit an airport, a railway station and bridges.

Iran hit an oil facility in Kuwait as well as a power and water plant, authorities in the Gulf state said, while in Bahrain the army said air defences repelled a wave of Iranian attacks.

Tehran also launched fresh strikes in Jordan, where the US military's Central Command said two service members were killed on Friday as they "defended against Iranian ballistic missile and drone attacks". It said another service member was still missing in action.

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Iranian supreme leader Mojtaba Khamenei, who took over from his father after he was killed in the opening salvo of US-Israeli strikes on February 28, said the ongoing attacks on his country "once again demonstrated to everyone the worthlessness of the American president's signature".

"Now that the American enemy seeks to incite war and bear its most serious consequences, it should know that the dear Iranian nation and the axis of resistance have unforgettable lessons to offer it," he added, in a statement carried by state TV.

Major General Mohsen Rezaei, a senior military adviser to Khamenei, warned that Tehran would resume "full-scale offensive operations" if US strikes continued in the coming days.

"Iran will no longer limit itself to retaliatory, like-for-like responses," Mohsen said, according to state media.