Natalie said:
I'm almost considering using a tough brush killer on them, even though I HATE using any kind of chemicals. We had to use some on the wild blackberries along one side of the property last year, because they were starting to take over. Huge areas of land here are totally covered in them, and we didn't want that to happen to us. So, it was chemical warfare, and it worked. I hope it doesn't come to that for the raspberries, but I'll do it if needed. This isn't an area that will ever have food planted in it again, so that makes the decision easier.
One thing you can do to limit the collateral damage caused by herbicides: Find the biggest, toughest plastic garbage bags you can, like "contractor bags". Pick out berry vine branches that are as far as possible from plants you want to keep.
Tuck the leading tips of some of those "isolated" black raspberry branches into it, then push them the rest of way in until you have stuffed the bag more than half-full. THEN spray some harsh but non-persistent herbicide into the bag, so NONE sprays around outside. Let the leaves absorb the herbicide. Leave them in the bag until any remaining herbicide dries out.
Then remove the bag and remember that toxin may still wash off the berries and onto other plants.
You might even leave them in the bags for days, until they've absorbed as much as they ever will, and have started to die back to the roots. Then cut them off from the rest of the plant while still in the bag, and throw them away.
If you know that all the herbicides in the mix have short lives, you might be able to try composting the dieing branches, but my guess is that blackberries can root even in their afterlife!