I'll say yes, they must be made of wood. Because not even all trees can produce wood that can make wands.
Only a minority of trees can produce wand quality wood (just as a minority of humans can produce magic). It takes years of experience to tell which ones have the gift, although the job is made easier if Bowtruckles are found nesting in the leaves, as they never inhabit mundane trees. The following notes on various wand woods should be regarded very much as a starting point, for this is the study of a lifetime, and I continue to learn with every wand I make and match.
pottermore article on wand woods
The article gives a breakdown of all the materials used to make wands and their various quirks and there is no mention of any non wooden material.
Furthermore, the pottermore article on wands states
Made from various kinds of wood, containing a core from substances such as dragon heartstring, unicorn hair or phoenix feather
pottermore article on wands
Harry Potter Wikia has a list named wands. And they're all wooden.
Voldemort's wand (according to the above article) has a bone handle in the film.