Answer: There are several weeds commonly called bindweed that meet this description and are very difficult to control without using chemicals. The best you can do is keep digging it out by the roots and cutting it off to try to exhaust the root reserves. Sometimes it can be smothered -- at leat temporarily -- under a layer of cardboard topped with organic mulch.
Here is a photo of one of the worst
I wish I could be more encouraging.
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