How to kill one plant next to others?

only by smothering it out with heavy black plastic

Best thing try to dig up

In keeping with Pung's idea, I would cut the plant as close to the roots as you can and spray the exposed stems with plant killer (weed killer, whatever the gardening dept. recommends is best) on a non-windy day and wait. Repeat as necessary.

HI Kitty. First cut the plant back as far to the ground as you can. Cut a piece of black plastic, or fabric that will fit over the plant and then put it over the plant for a few weeks. It should kill it off.

Going further on Pung and Seethebeauty's idea, paint the killer on the exposed freshly cut stem. That way you know for sure that no spray will get on any of your other plants and will only kill the sweet pea vine down to the roots. If you still notice sprouts, just repeat it, some plants are tenacious and it may take more than one try.

Here is an article that might help - Kill Plants

I hope it helps - good luck :)