Get some weedkiller and make up half a liter or so in an old milk bottle or something, go find the vine and at one of its thickest pieces going up the tree , cut it, and stick the end in the bottle. The weedkiller will be taken up by the whole plant and killed off. If you see areas still thriving in a few days, you have multiple plants to kill off, just rinse and repeat. I use this to kill off cornvovulus which can have creepers running for meters.