I don't think that the
Barrow-wights perished, though there might not be a way to know. This is because as I understood the text, the barrow wights were not simply bodies kept undead by some spell of the
Witch-king, but rather inhabited by some "third party"
spirits. In Appendix A it says that
That certainly sounds like the "evil spirits" acted quite independently from the Witch-king and were not necessarily created by him or even directly commanded by him. In Unfinished
Tales, The Hunt for
the Ring, it is stated in an alternative version that
Here it is made clear that the Witch-king actually sent the Barrow-wights to the Barrow-downs (so many hyphens

), but still they seem to be separate entities from himself.
Note also the poem
Tom Bombadil uses to rescue the
Hobbits from "their" wight:
Get out, you old Wight! Vanish in the sunlight!
Shrivel like the cold mist, like the winds go wailing,
Out into the barren lands far beyond the mountains!
Come never here again! Leave your barrow empty!
Lost and forgotten be, darker than the darkness,
Where gates stand for ever shut, till the world is mended.
Especially the last two lines certainly sound to me as if
Tom were not merely banishing the wight from that particular barrow, but "pushing" him outside the world through the Doors of Night like
Melkor at the end of the
First Age. This would suggest that the Barrow-wight was actually a minor Maia-spirit.
That last part is really only speculation, but I think it is quite clear that the wights were spirits independent from the Witch-king and therefore not bound to him like for example the foundations of
Barad-Dur were bound to
the One Ring. So I see no reason why they should have perished at his death.