“A fool criminal can come with one hit of a hammer and destroy all our efforts, and we can do nothing,” said Qais Hussein Rashid, deputy minister for tourism and antiquities. “It’s a great grief.”

Some have even called for airstrikes, not the usual province of Iraq’s cultural elite. Mr. Rashid and his boss, the minister, Adel Shirshab, both called for American-led coalition warplanes to strike militants approaching other historic sites.

On Sunday, the officials took their latest step in seeking designation of the ruins of ancient Babylon as a Unesco world heritage site, hoping for a measure of protection by the United Nations.

Yet the prospect feels like thin armor given the damage wrought to other Unesco-designated sites, like Hatra in Iraq, and, the Krak des Chevaliers crusader castle and the Old City of Aleppo in Syria. Those Syrian sites are victims not of the Islamic State, but of four years of conflict between government and opposition forces, who shelled them and used them for cover.

Now, Iraq’s cultural institutions are “on the front lines against terrorism,” Mr. Shirshab said, fighting a “barbarian invasion that is targeting our heritage.”

But Iraq, he said, has survived “many invaders.”

He was not referring only to Hulagu Khan, the Mongol conqueror who razed the world’s greatest library and some of its finest buildings when he sacked Baghdad in 1258. (At the time, Baghdad was the seat of the Islamic caliphate, while today the Islamic State is merely a self-declared caliphate.)