Are you sure it's ground cherry? That will die come winter in your climate. Could you post a picture so we know which plant is under discussion?

On further thought I wonder if you have Chinese Lanterns, Physalis alkekengi var. franchetii, rather than ground cherries.

I have ground cherry,

Down turned yellow flowers with a black center which then turns into a dull colored papery "lantern". Chinese lantern plant is a distant relative, but no that is not what I have. It's a big genus and I do not have the hairy leaved version.

No current pictures as I spent this morning digging out what I could.

The only ground cherries I ever grew pulled out quite easily so I can’t help further. Do yours regrow from broken off roots? Are they perennial in your climate?

If yours pulls out easily, it may not be Physalis. Mine is probably Physalis heterophylla

I just googled it, and it sound like a disaster, besides being poisonous. And yes, it does grow from pieces of root, like bindweed. You could try digging it up carefully with a garden fork, which doesn't break the roots like a shovel does. I have done that for mugwort, and it worked. If it is too monumental a spread to do that, try Ken's trick with RoundUp - put straight RU in a small squeeze bottle, then cut each plant almost to the ground, and put a drop or so of RU on the freshly-cut stem. That should at least greatly reduce the spread. You will probably have to do it again, to get the few that will resprout. I wish you all the good luck; it sounds dreadful. Not all native plants belong in the garden, unfortunately.

lisanti, the squeeze bottle of RU is a great idea! Because these horrors almost always are right next to plants I don't want to destroy, a squeeze bottle will let me target them without killing anything else.

By straight RU do you mean the concentrate or do you just mean regular RU with nothing added to it.

Mine were Physalis peruviana, an edible fruit crop, although several other Physalis are called ground cherries or tomatillos.

Peruviana is definitely an annual in these parts. Apparently the common name is cape gooseberry not ground cherry. The ground cherry also produces edible fruit but I have no desire to try it all I want is for it to be gone!

The concentrate, which is actually something like 41%. To reduce the threat to nearby plants, how about putting the RU in a jar/can (that you don't tip over!) and using a small paint brush to get the RU on the stems of the Wicked Invader? Our own Ken from Adrian puts some motor oil in the RU to make it thicker; not having motor oil, I have used dish detergent to thicken it up some.