Only had my car a few days (driven <100 miles) when I got caught by this.
So, I am 20 miles from home. The car starts to die. Narrow two-lane road. Car behind. The rear wheels chirp! I coast to the side of the road, luckily finding a turn on section at a road junction. Engine stalled light on. Restarts badly and briefly. Hazards on.
I am THAT guy.
I had noticed the fuel gauge was not going down. But didn't think about that enough! I call my wife to bring out a can of 93 gas. Meanwhile I read up on my phone and convince myself that the engine has blown and I have a wreck to explain to my wife. I was almost literally shaking when she arrived. So, I stand in the road, dodging cars and trucks as I fuel the car. Started up fine!
A few minutes later I filled up at a gas station to the pump's click (taking out an extra mortgage to do so). And it has been fine since.
I noticed that the fuel gauge did not go to full.
Presumably the fuel tank sender unit (float) has broken?
I have read that this is a pig to replace. Tank out, etc.
Any advice?
I can sort of live with it, but I am a bit of a freak for details. A faulty gauge reading might eventually put me in an asylum!

 

The pump can is a bear to change. But the level sender isn't so bad. Unlike a lot of fuel pumps where it's integrated into the pump, on the Jag it's mounted on the "back" of the tank, as it were. If you remove the trunk lining and remove the metal plate covering the tank you'll see a round thing. That's the sending unit. You can unplug the wires, unscrew it and pull it out. Replacement is the reverse. Be sure to do this with less than about 1/4 tank of gas.