Can any dead become a wight or must they be victims of the wights or White Walkers? The question comes from a discussion around people, horses and dragons being reanimated.
Your Grace, if my father had seen the things that I've seen, he'd also tell you to burn the dead before nightfall.
-- Jon Snow's words to Stannis after Stannis' troops attacked a group of wildlings and killed many of them. Those wildlings were not killed by White Walkers of wights, and yet Snow - who by now knows what he's dealing with - is clearly implying that they can be raised.
Judging by this, you do not need to be killed by wights or White Walkers to become a wight.
Additionally, when Meera, Jojen, Hodor, and Bran reach their destination at the weirwood tree, they are attacked by several wights and first meet Leaf, a Child of the Forest. Jojen is stabbed multiple times by the wights, and is then killed - not by wights, but by Meera, who cuts his throat to spare him a much worse death at the wights' hands. They then follow Leaf - but, before they leave, Leaf makes sure to burn Jojen's body with a fireball.
From what we've seen on the show, I haven't read the books, it would seem that any creature can become a wight if they die in the area of influence of whatever magic that causes it.
Said area was limited by the Wall, but now that it has been breached, it seems to expand as the Army of the Dead advances, as proved by the transformation of the Umber boy at The Last Hearth.
Also, the wildlings had the habit of burning their dead, and it's highly unlikely that they all die at the hand of a wight or a White Walker.