My husband's XJL (2014) also stopped working. It took quite a search, but he was able to find the battery for it. I believe it even had to come from Germany or England. A very specific battery. His car had closer to 50,000 when it stopped.
Yes, it is a wholly separate battery - maybe 1/3 the size of a normal battery.Mine too has stopped. Research tells me the separate battery that operates this feature needs replacement. It is located under the carpet in the trunk. I haven't replaced it yet but intend to.