glyphosate based weedkiller brushed on. Although brambles are a bit tougher if established than the average weed, so SBK brushwood killer. However this must only be applied to the brambles.

May I suggest waiting until the winter when you can see what you are doing and many other plants will have died back too. Also you can enjoy the final crop of blackberries.

SBK's the way to go, but don't spray it on the foliage or water it on. Cut the brambles right back to the central woody root in the ground- dig around the root a bit to clear the soil away and if you can't dig it out, drill into it with a medium wood bit to make a couple of holes about quarter inch deep, then pour in neat SBK - wear gloves, don't spill it anywhere else at all, not even the soil, and be careful because its like water. Then cover the root with something, an upturned pot or a stone or a plastic bag, anything, shove some soil around and leave it to work.

You can do it without weedkiller. I cut the growth back to about 3 - 4 inches, then grub out as much of the roots as possible. When there's some regrowth in a few weeks the remaining roots are shorter (I have a theory that the plant reabsorbs some of the root to produce new shoots and leaves) and are much easier to get out. I cleared an extensive bramble patch last May / June and while I have to pull up bramble seedlings (easy peasy) the roots seem to have all given up.