Killing Austrian Winter Peas without herbicide?

I think this is really two questions in one.

First, would you expect Austrian Winter Pea to be winter killed in Zone 6? Maybe I chose a bad cover crop.

Second, now that these things weren't winter killed, what should I do to kill them? I did cut some off at ground level, but it looks like they have leaves all the way down so I'm wary that they will grow back.

Thanks!