A recent article in Current Anthropology describes re-analysis of textile fragments from Guitarrero Cave. Guitarrero (which despite my best efforts I persist in spelling Guitarerro) is a rockshelter in the inter-mountain region of Ancash province in Peru. The cave has evidence of human occupation back to 12,100 years ago, and in the cave were found cords and textile fragments direct-dated to that occupation.
Both sides of a fragment of a woven mat or basket container from Guitarrero Cave. Black grimy residue and wear from use is visible. Photo © Edward Jolie and Phil Geib
The textile fragments show a z-twist design and are made from agave and bromelid fibers, plants which are today found outside the cave opening. Guitarrero cave is the oldest to date example of textiles in the Americas. However, it's most likely that the technology was brought to the Americas with the first colonists, called Pre-Clovis, since using textiles is something we humans have been doing for some 30,000 years.
Jolie EA, Lynch TF, Geib PR, and Adovasio JM. 2011. Cordage, Textiles, and the Late Pleistocene Peopling of the Andes. Current Anthropology 52(2):285-296.


Comments
I wonder if pre-Clovis people brought weaving with them? Was it practiced in the Arctic/sub Arctic regions they went through? Or, did they (re)invent it after reaching warmer climes with plants suitable for providing fiber? Interesting question.
The Old Gringo
Is a woven basket considered textile?
Does weaving a basket or mat necessarily translate into weaving cloth for clothing or blankets?
Many of the native American peoples, before contact with Europeans, wove baskets. But still wore animal skin clothing and used animal skins for blankets. And did not weave cloth.