Probably not
In A Feast for Crows, Chapter 5 Samwell Tarly tells Jon Snow that he found one account of the Long Night that spoke of the last hero slaying Others with a blade of dragonsteel and they could not stand against it and Jon wonders if dragonsteel refers to Valyrian steel.
“What about the Others?”
“I found mention of dragonglass. The children of the forest used to give the Night’s Watch a hundred obsidian daggers every year, during the Age of Heroes. The Others come when it is cold, most of the tales agree. Or else it gets cold when they come. Sometimes they appear during snowstorms and melt away when the skies clear. They hide from the light of the sun and emerge by night... or else night falls when they emerge. Some stories speak of them riding the corpses of dead animals. Bears, direwolves, mammoths, horses, it makes no matter, so long as the beast is dead. The one that killed Small Paul was riding a dead horse, so that part’s plainly true. Some accounts speak of giant ice spiders too. I don’t know what those are. Men who fall in battle against the Others must be burned, or else the dead will rise again as their thralls.”
“We knew all this. The question is, how do we fight them?”
“The armor of the Others is proof against most ordinary blades, if the tales can be believed,” said Sam, “and their own swords are so cold they shatter steel. Fire will dismay them, though, and they are vulnerable to obsidian.” He remembered the one he had faced in the haunted forest, and how it had seemed to melt away when he stabbed it with the dragonglass dagger Jon had made for him. “I found one account of the Long Night that spoke of the last hero slaying Others with a blade of dragonsteel. Supposedly they could not stand against it.”
“Dragonsteel?” Jon frowned. “Valyrian steel?”
“That was my first thought as well.”
So while we have no definitive answer it seems very likely that at least the Last Hero of the First Men used Valyrian steel to kill a white walker.