You can win but it takes time, patience and persistence! The roots run everywhere, creating new plants on the way and this give it great resilience because the plants are interconnected and it can get food from its support network, even if some plants cannot photosynthesise.
What works for me is this.
First loosen the soil with a large fork - the roots go deep. Then starting from a green bit, track each and every root as far as you can, and remove it. Usually it will connect with several other plants, giving you new avenues to follow - a bit like a good detective story! Make sure you get the purply centre of each plant, but if you can't, because it has grown in the middle of a clump of something you want, then remove every bit of leaf you can. If there are stones, rocks, logs or paving flags nearby, lift them and look underneath. You will likely find a mesh of white roots like spaghetti, so get them out! It can feel very satisfying to track them all down.
When you have cleared the area, leave it for a week or two. After this you will be able to see new shoots from bits you missed - almost inevitable, because every tiny bit of root left behind can make a new plant. Get rid of all of them and fork the soil over thoroughly.
After this just keep a watchful eye on your clean patch and remove any new plants you spot straight away. Do not let plants flower and set seed in your garden and check boundaries in case it is coming in from outside. If all else fails you can eat it. Haven't tried it myself but my sheep quite like it
I spent a lot of lockdown last year pulling up ground elder roots. As @Buttercupdays says, it can be immensely satisfying. And am resigned to having to dig it up again and again as it really does root from the tiniest bit of root. The roots have a pleasant smell and yes, you can eat the young leaves. We tried it in salad.
You need a kneeling pad, a hand fork, a transistor radio, slightly damp soil, a bucket for collecting the dug up roots and hours and hours of time. My ground elder is virtually eliminated.
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I'll try lay down some black polythene it and see if it will get rid of it eventually as suggested on the site.