You could cut them, scrape the thorns off, split into narrow strips, soak in water and weave them into a quiver.
If you don't already know blackberries are biennial, meaning they grow vegetatively the first year and flower and fruit the second so if you cut them all down you won't have blackberries next year. Only cut the canes that fruited this year.
I'd bet you could find red osier dogwood somewhere near you and it makes great arrow shafting. Maybe one of the viburnums too.