Not entirely but bits and pieces were used in armour construction for the natives of Siberia and Alaska.

Wooden greaves (shin guards) were mentioned in the 6th Century Greek war manual Strategikion but it doesn't say how they were constructed or how effective they were. **Note, I have not personally read this or confirmed this to be true! **

Early kendo bogu were made from bamboo slats, if that counts.

For the most part, cloth armour was cheap, protective and repairable whilst wood is not.