Hi Fatpoppy and welcome to GoY. Weed killer isn't going to touch raspberry cane runners. Cut them down to about 6 inches above the ground then don a pair of good solid gardening gloves and pull and keep pulling, alternatively after you have cut back to 6 inches dig the root out back to the fence line. If you keep cutting and pulling, digging up the roots you will slow them down but not stop them. You could always grow raspberries and enjoy the fruit!

I think that you are going to have to dig them out, FP, and then watch out for them at your boundary.

Dig a small trench along the boundary and put some folded compost bags in, Push up to the boundary with the soil from the trench. It should be about 6" - 8" deep. Raspebrry runners don't tend to go lower than this. Then dig out what you have. On the bright side you could sell them back to your neighbour as year old runners. :-)

Brilliant 2ndhand. I was going to get some membrane, but compost bags are free - I need to stop mine invading the gooseberries.