Best you can do is to remove the spiny growth right back to its point of origin and hope.

If the desirable part of the tree is still alive, it should recover. You'll have to keep an eye on it though, and rub out any new suckers while they are still soft.

As above, you need to act quickly to remove the root stock suckers or it'll kill the plum

Thanks a lot for the responses. The desirable part of the tree is ok, just hasn't grown very much. I have cut down the errant one and will keep an eye on it from now on. Any ideas on what they use as root stock? The leafs looked like the plum but about half the size. The spines were about two inches long, haven't seen spines like that on anything other than cactus. Would wrapping the stump with something stop the suckers from appearing?

Often Quince is used as a root stock

Stopping them I suspect is like apples, just keep pulling them off, ideally as close to the root as you can get without disturbing the root & triggering even more

Quince wouldn't work for plums, but wild damson or sand cherry would.