That's a veneer panel that is so thin that I can't imagine it was held in place by nails. I suspect that whatever the glue was it was insufficient or has failed. To fix it, first take the panel completely off, since it looks like at least half of it has come away and is being partially held on by the electrical plate. Then you would need to buy good quality carpenter's glue like Titebond and spread it evenly with a roller and position the panel back in place. The hard part for you is that it does need to be clamped, and I suspect that you don't have much in the way of specialized clamps to do this. If you have doors on both sides of the island, you could temporarily remove them, cut a piece of 3/4" plywood to the size of the panel, and then clamp this up against the panel while the glue dries. If you don't have doors on both sides, there are other ways to do clamping, but it's more complicated and I'm not going to try to detail it here. If this sounds like it exceeds your skills and available supply of clamps, then you need to get a handyman or cabinet person in to do the repair.

If there isn't an easy way to clamp it, you could use glue and 23 ga pin nails.

You can make some inexpensive clamps using PVC drain pipe and a hacksaw. Might work if the cabinet face isn't too wide. I think I saw this first on one of Joe Corlett's pics.

How about wrapping the island with ratchet straps to clamp the veneer ?

I had a piece of paneling come loose and I used Gorilla Glue construction adhesive. I sat on the floor and used my feet to hold it in place, lol. It didn't take much longer than 10 mins for it to stick. (it was the easiest way to do it)