Looking for a service manual. Any clue where I could acquire one without having to give up an arm and a leg?

 

Not sure where you live so that might matter. But EBay has a number of discs and the price is low (Under $20). Now understand the JLR manuals are HUGE! I mean like 8000+ pages. They are also very repetitive so the same thing is printed on page after page. Next you will see Diesel and gas manuals so find one for your drive train. But for the price they are worthwhile.
Jaguar XF Workshop Manuals

The gold standard is to subscribe to the JLR Topix. They charge by the minute. Plus they are still recovering from a hack of some kind and my log in has not worked for months.
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I ordered from easymanuals.co.uk
Not bad but as it is a PDF the images can only be resized so far before they become blocky. They have different manuals for year and engine size/type