WASHINGTON/BAGHDAD, Jan 20 (Reuters) - Missile attacks in Syria, Lebanon, Iraq and Yemen on Saturday threw into sharp focus the increasing risk of the war in Gaza triggering a wider regional conflict pitting Iran and its allies against Israel and the United States.

Iran said five of its Revolutionary Guards were killed in a missile strike on a house in Damascus for which it blamed Israel, and security sources in Lebanon said an Israeli strike there killed a member of Hezbollah.

Later on Saturday an American official said a missile or rocket attack had injured U.S. personnel at a base in Iraq, a country where Iran-backed groups have previously battled U.S. forces.

The United States also said it had targeted a missile the Iran-backed Houthi group in Yemen was aiming into the Red Sea, which it called a threat to shipping.

The conflict in Gaza began on Oct. 7 when Hamas fighters stormed border defences to attack Israeli bases and towns, killing more than 1,200 people and seizing more than 250 hostages.

Hamas is part of Iran's "Axis of Resistance", a regional alliance that also includes Lebanon's Hezbollah, the Syrian government of President Bashar al-Assad, Shi'ite militia groups in Iraq and the Houthis who control much of Yemen.

Israel's intense bombardment of Gaza since Oct. 7, which health authorities in the Hamas-run enclave say has killed nearly 25,000 Palestinians, quickly triggered border clashes between Israel and Hezbollah.

Over the past three months Israel has also repeatedly struck at Iranian targets in Syria, while Iranian-backed groups in Syria and Iraq have fired at U.S. targets in those countries.

Aside from Gaza, the theatre of conflict with the widest international repercussions has been the Red Sea, where the Houthis have repeatedly targeted shipping they say is bound for or linked to Israel, hitting global trade as some companies avoid the key waterway.

U.S. and British strikes over the past week have targeted Houthis forces in Yemen.

Regional and Iranian sources say Iran and Hezbollah have personnel in Yemen helping direct attacks on shipping, though the Houthis have denied that.

(Writing by Angus McDowall; additional reporting by Laila Bassam, Maya Gebeily and Kinda Makieh; editing by Giles Elgood)