The oldest known instance of textiles survived as an imprint on clay only — these were reed matting or possibly baskets which left a beautiful clear impression on wet clay, which then dried and preserved the image. This was found in Europe somewhere, is circa 30K years old.
Oh, okay, here are some 30K year old flax fibers, from Georgia (in Europe, not US):
http://www.npr.org/templates/story/story.php?storyId=112726804
Interesting. I was hoping the opening article would say what kind of plant was used in the Peruvian textiles.