Salmonberries are persistent! I actually use them as part of a process of reducing/eliminating my Himalayan blackberries in my "permaculture zone 3" area. I like salmonberries more than blackberries, but less than thimbleberries or raspberries, so I encourage the salmonberries until they mostly eliminate the blackberries, then encourage the thimbleberries until they outcompete the salmonberries. This actually works out pretty well, and takes less time than digging up every root.
(Here's my "order of operations" for moving from plants I hate, to plants I like: bindweed->evergreen blackberry->Himalayan blackberry->salmonberries/trailing blackberries->thimbleberries/blackcap raspberries/domestic raspberries).
But, this takes a few years, and I wouldn't suggest this kind of operation for a
garden bed (it's better for converting it from one type of hedge to another). When I wanted to turn a trailing blackberry/salmonberry patch into my son's garden bed, I tried to remove as many roots as I could, and then layered paper
feed sacks and even some old cotton shirts to make a barrier, and then covered that with inches of duckbedding, and then put potting soil on top. This actually worked out okay. Some of the blackberries came up (I'm sure if they were Himalayan, even more would have come up). But, like Kim said, they are easier to pull, and often come right out. I did have to do quite a bit of weeding, and during the hight of summer, I would visit the garden bed at least twice a week to pull out tiny little blackberry shoots. That was last year. I found about three so far this year, but they were really tiny and easy to pull up. It's also hard to tell how many are migrating in from other areas. But, I managed to have a garden that grew really well the first year, and probably would have done even better had I done a double layer of material to smother the ground.
In another area, I wanted to make a nice place for my daughter's pear tree. The intended area was all salmonberry and invasive blackberry. Since I knew I'd be planting the tree on her first birthday, I started prepping the area while I was still pregnant. I first cut down all the salmonberry and blackberry as close to the ground as possible. Then I layered feed sacks and covered with about 4-6 inches of woodchips. I noticed some salmonberries and blackberries popping up through, so I clipped those short and put another layer of feedsacks on top of the whole mulched area and then added another 2-4 inches of mulch on top of that. About 6 months after that, I planted my daughter's tree. It's been another 5 months since then, and so far, no more blackberries or salmonberries have popped up. We'll see what happens when spring and summer are in full force, though!
The area I mulched for my daughter's tree (
thread:
https://permies.com/t/69253/Pear-Tree-Daughter-Birthday):
Here's my son's garden, with a before and two afters (thread:
https://permies.com/t/58238/critters/garden-soil-poultry-poop-pine):