No, perhaps threatened at some point like so many other whale species, but they have never been listed as extinct. The tusk is thought to have been one of the inspirations of those stories of the fable animal the unicorn
Intriguingly, there is another variety of unicorns in folktales and old poems of Central Asia and China, a more robust, angry and furry one, and it's been speculated that the eastern variety would have been inspired by another animal that actually went extinct at some point in the later part of the last ice age or even a few millennia after it ended - Elasmotherium
, a kind of big woodland/steppe grazer looking like a rhino but more than twice as large on every axis and with a big single horn. Nobody's been able to make sure just when this species of beast breathed its last, and some scientists think it survived in small isolated groups to somewhere near the dawn of recorded history or even later. The early migrants heading towards the Bering land bridge region and later on into America could have been familiar with it, too (no signs of the animal ever living in North America though, but any remains of it in Canada or New England would have been razed by the last glaciation).