You can dig it out thoroughly in the spring although it is difficult to get out every piece at the first or second go. Or you could spray it with glyphosate ,taking care not to get the spray on other plants .

one bindweed plant can give the impression there's lots more of it than you think. If you can get right down to ground level and have a look. you may find there are far fewer plants than you thought and you can whip them off which will at least stop the top growth setting seed. In a few days when the top stems die they will be alot more obvious and you can pull them out.

The roots go down several feet and are very brittle - not for nothing are the called "devil's guts". Pulling the tops off every time they appear will eventually weaken it but you have to be very vigilant and it takes quite a while. Another tack is to put in some canes and train the bindweed up them. Then when its quite big lay the canes flat, protect surrounding plants and spray with glyphosate. this makes spraying a lot easier than trying to do it when the vines are twisted round other plants.

Cut the bottom off plastic eg Coke containers. Unravel the twined leaves if you can. Loosely insert into to the bottle and skewer to soil with a small support cane. Squirt Roundup or similar through neck of bottle, before inserting the cane. Otherwise, as suggested, just keep persistently pulling off the growing tip. See who gives up first!!!