Kudzu makes great baskets, all the more reason to harvest it.
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For better or worse, it is not native to this area. I saw a lot of it in Arkansas, which is where I learned, first hand, about the smell...
I'm going to assume that when it dries, the smell is greatly reduced, because I would never want food stored in a basket that stunk like Kudzu.
Locally, we do have a few climbing vine plants, but I've not looked into basket making using local plants.