Kuwait's 346,000 b/d Mina al-Ahmadi refinery came under new Iranian drone attacks in the early hours of 20 March, state-owned Kuwait Petroleum Corporation (KPC) said.
The attacks caused a fire at several units of the refinery, but no injuries were reported, a KPC statement said.
Emergency response teams are in the area and have been trying to contain the fire.
"As a precautionary measure, a number of refinery units were shut down," KPC said.
The new attack comes soon after Kuwaiti and Saudi Arabian energy complexes were damaged from drone hits on 19 March.
Iran earlier warned that retaliatory attacks in response to the Israeli bombing of its gas processing facilities will continue.
US president Donald Trump said Israel agreed to his request to stop targeting of Iran's energy facilities.
Israel's prime minister Benjamin Netanyahu later confirmed his agreement to hold off attacks on Iranian gas fields.

