Depends on the variety. Erect varieties do, trailing varieties do not. Sounds like yours is an erect/semi erect variety which will send out runners.

The wild blackberries around my yard most definitely do, as if I go a couple weeks without mowing the yard I will see blackberry vines cropping up upwards of a couple feet away from the vines proper.

I know mine would tip into the ground and root from there...don't recall them spreading like raspberries.