The oldest surviving basket was allegedly found near the Dead Sea, and carbon dated to 10,500 years old; It is of advanced design, large and with a lid, so basketry had clearly been long established at the time it was made.
Unfortunately, as this artfact was allegedly found with/near the Dead Sea Scrolls, many people have a strong motive to lie about it, so yet again, religion ruins everything - there's no compelling reason to doubt its existence or its age, but it is conveniently being used by lying liars who lie to support their unrelated claims, which sets my skepticism alarms jangling.
It would be surprising to me if basketry didn't have a similar antiquity with flint knapping, both of which are complex manual skills whose products are highly useful, but which require only fairly ubiquitous raw materials and can be made without precursor tools of any kind.
Obviously flint knapping leaves far more durable evidence, but I would be surprised if basket weaving wasn't slightly older.