Without a photo it is difficult but it sounds like bindweed to me - yuk! Are the leaves heart shaped and there are white flowers later? You'll never get it by digging as the main root, which runs horizontally in the soil, can be as deep as 2 metres, but mainly it's higher than that. If you don't mind using chemicals then I would paint some weedkill gel on the leaves everywhere you can and wait.
Some old gardener once told me that if you dig out the blighter as soon as it pops out of the ground and make sure you get every shoot then eventually it gives up and dies but if you let just one shoot go and the leaves get the sun then it stays.
Luckily I don't have bindweed in my present garden but I have battled with it in the past - and won. Good luck.