The “rabbits ears” and flat pads are stems.
Yes, it’s easily rescuable. Firstly, take it out of the pot. Does it still have healthy looking roots? If so, re-pot in a gritty compost and follow all of Nan’s advice.
If not, then detach the basal pad. Make up a pot of gritty compost, leave the rest lying on top-of the pot. It should start rooting at the base. you can then pot it up. Don’t bury the pad, so you’ll need to prop it up. It’ll be horribly top heavy so you could separate all the pads and propagate them all separately.
You can also take the basal pad, cut it above any rot, leave it lying for a while until the cut edge has calloused over, then prop it up on a pot of gritty soil, and it will likely root.
These things propagate themselves by falling over and rooting from the nodes between the pads. They are basically invasive weeds in the right climate. So you have a very good chance of success.
From September onwards this needs very little water (if indoors) or none at all if ina porch, conservatory or greenhouse. It can stand cold provided it’s bone dry. There used to be a couple of collectors, the Lambs, who had the Exotic Collection near Worthing - all their Opuntias were outside over winter, but shielded from the rain. I’m not recommending you try this, just illustrating that it’s the wet that kills them, not the cold.