US Defense Secretary Mark Esper on Sunday said air strikes against a pro-Iran militant group in Iraq and Syria succeeded, and he did not rule out additional action "as necessary."

"The strikes were successful. The pilots and aircraft returned back to base safely," Esper told reporters after F-15 jet fighters hit five targets associated with Kata'ib Hezbollah in Western Iraq and Eastern Syria.

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The targets were either command and control facilities or weapons caches for the militia group, he said, hours after the Pentagon announced the strikes.

The US action followed a barrage of 30 or more rockets fired on Friday at the K1 Iraqi military base in Kirkuk, an oil-rich region north of Baghdad. That salvo killed a US civilian contractor and wounded four US service members as well as Iraqi security forces.

Esper said that he and Secretary of State Mike Pompeo had travelled to Florida, where President Donald Trump has been spending the Christmas holidays, to brief him on the latest Middle East events.

"We discussed with him other options that are available," Esper said. "And I would note also that we will take additional actions as necessary to ensure that we act in our own self-defense and we deter further bad behavior from militia groups or from Iran."