Unfortunately, there isn't a simple solution to controlling wild blackberries without herbicides. It will take multiple techniques and regular checking to eventually remove the blackberries from your plot.

Generally, wild blackberries will resprout from the rhizome just at or below the soil line. There is a small possibility that the canes can resprout and/or re-root depending on where they were cut off and if conditions are right. Removing the canes and allowing them to break down off-site would be a good idea. Then you will want to stay vigilant to check the plot for resprouting and destroy any new sprouts/foliage that emerges. Eventually, the root system will 'starve' and die. It may take several years to completely remove blackberries from an area.

You may get frustrated pretty quickly trying to dig out the roots. Blackberries have an extensive root system that is a lot of work to completely dig out. Be aware that if you break up the roots that each root fragment can resprout and start a new plant. So you might see a flush of new plants that will need to be tilled or dug out later in the growing season.

There isn't good research on the use of black plastic to smother wild blackberries. This technique might work to kill off the foliage but wouldn't negatively affect the root system.