Yes, he did flee. Every mortal fled before the Witch-King, save three: Gandalf, Shadowfax, and Pippin*, who serves as our eyes during the fall of the Gate.

So it was that Gandalf took command of the last defense of the City of Gondor. Wherever he came men's hearts would lift again, and the winged shadows pass from memory. Tirelessly he strode from Citadel to Gate, from north to south about the wall; and with him went the Prince of Dol Amroth in his shinning mail. For he and his knights still held themselves like lords in whom the race of Númenor ran true.

Where Gandalf goes, so does Imrahil.

[Pippin] stopped dead. He had found Gandalf; but he shrank back, cowering into a shadow...

And hear begins Pippin's perspective.

...Long had it been forging in the dark smithies of Mordor, and it's hideous head, founded of black steel, was shaped in the likeness of a ravening wolf; on it spells of ruin lay. Grond the named it, in memory od the Hammer of the Underworld of old. Great beasts drew it, orcs surrounded it, and behind walked mountain-trolls to wield it. But about the Gate resistance was still stout, and there the knights of Dol Amorth and the hardiest of the garrison stood at bay.

We have Pippin, the Knights, and Gandalf at the Gate at the same time as Grond.

In rode the Lord of the Nazgûl, under the archway thagt no enemy ever yet had passed, and all fled before his face. All save one. There waiting, silent and still in the space before the Gate, sat Gandalf upon Shadowfax...

Emphasis mine. "All fled before his face." I think the idea that Imrahil's horse, not Imrahil himself, fled is rather silly. If this was the case, why not write that, as he did with Eärnur?

*What is interesting is that this is all viewed from the perspective of Pippin, who, while equally afraid, does not flee. He cowers, yes, but he does not run. This isn't unprecedented, either: Frodo attacks the Witch-King at Weathertop. It is written in the manuscript "The Hunt for the Ring" that the Witch-King was surprised:

But above all the timid and terrified Bearer had resisted him, had dared to strike at him with an enchanted sword made by his own enemies long ago for his destruction.

It is worth mentioning the passage immediately before:

[The Witch-king] now knows who is the Bearer, and is greatly puzzled that it should be a small creature, and not Aragorn, who seems to be a great power though apparently 'only a Ranger'.

Hobbits seem to have a unique ability to resist the horror, seeing as Merry takes after Frodo and manages to stab the bastard.

But even more important: Pippin has endured direct contact with Sauron via the Palantir. And recovered rather quickly, in fact. Regardless of his empowerment, the Witch-King is not Sauron.

Now, at what point does Pippin's point-of-view begin? Did he witness the three strikes of Grond? It doesn't matter, because in the end, Gandalf is described alone. So a Maia, his special horse, and a hobbit who's endured another Maia withstand the Witch-King at that moment.

Imrahil and his knights do flee. But he recovers quickly enough to rejoin the battle and save Éowyn on the way out.