Rent equipment and remove the entire patch or it will constantly seed and branch back into the new cleared area.

Do you happen to have a chipper? We cut vines into 3' lengths and run them through a chipper. Digging out the roots is good, but once they are gone the best way to keep them away is to mow. If you want to have a garden in the area then you will have to keep digging them out. I also don't like herbicides, but in the case of blackberry vines I have been dripping brush killer with triclopyr just on the cut surface of the vine close to the ground, in hopes of killing the roots. It seems to work at least in part and prevent their return at least for several years. It also helps with difficult resprouters like bigleaf maple or hazelnut trees.

Might be something useful here:Topic from 2011
Scroll down a ways on the tallcloverfarm link to get to the removal info.

What type of blackberry is in your native pockets? Only R. ursinus (dewberry) is native here. It can be invasive also, but on a daintier scale.

I greatly appreciate all of the wonderful advice! The link was also incredibly helpful. R. ursinus is the native on my property. It has taken over about 1/10 of an acre, but nowhere near the garden space, so I have let it be (for now). At least it is not Himalayan - that has laid claim to many acres. This year I am only hoping to manage the garden+immediate surrounding areas and now I have some hope :) I will be mowing the invasive down regularly and look forward to seeing some results! Thank you again everyone!

Another option is to smother with overlapping cardboard & heavily mulch with arborist wood chips 1-2' after cutting to the ground. You might get tree trimmers to dump them free at your place.

I have had super luck spraying with Crossbow. Seems to only kill brush!

The ursinus berries can be sparse, but are a very desirable fruit.

Wear lots of ankle protection, look for berries in July-August.