I used to get them on their migration northward. They loved our ground feeders and would hang around for several days before getting back on the road. So lovely...and they had no fear of our cats (who never harmed a bird, ever).
You can try keeping Rescue Remedy on hand. I used to keep a couple of bottles handy for the front yard and the back yard. I've used it on birds I was sure were dead, as well as a couple of squirrels who fell from great heights, a gopher (a tortoise here in Florida) and more. Except for the gopher who was alive, but only barely, all of them were dead still with no (discernible) heartbeat or respiration. A few drops of Rescue Remedy, some gentle stroking, a few more drops, continued stroking...and they all revived.
I have also used it twice to no avail, once on a bird and once on a squirrel. It's sure worth trying, though. Y'never know!