Probably the computer. Over or under pressure might trigger the computer to shut down the fuel pump. A belt slip, if not major would trigger the computer apply limp mode to protect the engine from internal damage.

Honestly, I am hoping, but not confident. It seems odd that if either were the problem it would take 4 to 10 hours to bleed-off. Also, I am still not clear on the dynamics of how a over or under pressure situation in the tank would cause the fuel to suddenly completely stop flowing. Or like with a belt slip, it potentially puts it in "limp" mode, does low fuel cause the computer to shut things down?

Any other ideas would be wonderful!!

Hope it does the job!

If it doesn't, maybe the gas cap is next... (if you haven't already).

New "gas tank breather valve" in place ... testing this week. Will report at end of week.

One thing I will say this was one of the easiest fixes I have done on the Mini Just remove the ground and it slips right out, easy to get to as well.

There is a gas tank breather valve in the engine compartment:

There is one OTHER thing I should have mentioned on this last failure. Was under a load, going up a pass (Sexton Summit) when it failed. The fuel pump was quiet (even with the seat out) ... BUT, when I pulled over I tried to start it, waited a few minutes, removed the gas cap and it restarted. Ran like crap for about 30 seconds before clearing up and running normally.

I didn't think of mentioning this since I have tried this before thinking it may have been a vacuum lock, BUT i may have been chasing a bunch of problems at the same time ... I have not noticed venting on this system ... is it all in the gas cap? It's stock so maybe that is next? 

I will say it did not make any noise when releasing the cap (no suck sound or blow out sound).

Thoughts.

I would love to ... I am running out of ideas. The problem is the closest mini dealer is 3 1/2 hours away from me ... and it's in Portland, which is kinda a mess right now

It might be time to take it to the MINI dealer....?

BOOOOOO!

Bad news ... it died again!! Good news was it started back up relatively quickly after stopping. 

replaced battery - failed (it was bad)
replaced serpentine belt - failed (it was bad)
Replaced Fuel filter and pump - fail

replaced fuel pressure regulator and housing for fuel filter - fail

NEXT???

Anyone in southern Oregon want to take this off my hands?? I am close to the give up point

Yes, good to know how the issue was solved for the next person experiencing similar issues.

Thanks for sharing.

We always appreciate follow-up posts once a solution is found or even attempted!

Everything is finally installed and running. Had a little problem with the new Fuel Pressure Regulator/Fuel Filter Housing. Since I had a new filter in the old system, I swapped that and apparently I broke the o-ring and it would not get fuel pressure. It took some thinking to figure that one out, but when we started to bleed off the fuel from the line, it was not coming out very strong, so figured the new part was bad or I had installed it wrong ... this time it was me.

Anyway 70 miles so far ... running good. Will keep all informed if things change.

Figured that since I put the old pump back in to check if I messed up the wiring for the float, without reconnecting the fuel lines, it showed me it was on a timer!

I'm late to this party /thread.

Here's my experience:

I had an 05 S with similar symptoms.

Replacing the fuel filter seemed like it helped for a while, then it came back. Replacing the fuel pump seemed like it helped, but then it came back. Replacing the coil seemed like it helped a while, then it came back. Found a damaged bladder valve thingie on the fuel rail that is labeled a fuel pressure regulator. I think I replaced that at the same time as the coil. Either way, it came back after replacing that as well.

What ended up fixing it is when I replaced the entire in-tank fuel filter side assembly. That is the actual fuel pressure regulator. It is a non-return system and (most fuel injected cars of the era have a second line running back the entire length of the car return excess fuel back to the tank. The MINI setup here just has it pressure release/dump right back into the passenger side of the tank after being pumped out of the driver side of the tank.

Hearing the fuel pump running and then cut off is a red-herring. It is on a timer. It will run when you open the door for a second or two, and it will run again when you put the key in the ignition and turn it. But that doesn't mean it is up to pressure when it cuts off. It just means it has stopped due to the timer telling the pump to "stop". If you experience it cut off again, when you pull over, cycle the key about 5-10 times to hear the pump fire up each time and then try cranking it and see if it restarts immediately.

That's what I thought as well based on what I see from ebay, when they sell the entire starting assembly it shows that part, otherwise it's listed as an ignition control unit is sold separately ... and wondered if this was failing would it sputter occasionally or would it be a hard fail? Also, is it tied to the chips in the keys, and if so could that easily be reprogrammed in the field?

I ran the car 100 miles with no problem, BUT have to remove the pump since the fuel float is not registering. On the bench, the float circuit is open except in around 1/4 tank where it's intermittent, so it's down again for a little ... I have a good hard backed manual and still cannot find that part yet. 

That looks like the EWS electronic immobilizer that is tied to the ignition. 

I believe it is located under the top part of the dash above / behind the steering wheel.

Seems to look like this, I don't know where its located. If you can get one cheap to carry with you, i guess you could just swap it out if the car dies. But that's nothing like testing, and has a good probability of not being the problem.

Ok, Installed a new pump and filter, the filter was very dirty. It is making less noise than the old one, so that is good.

The first long drive went well, hoping for the best, but still, need a few weeks before I am happy. 

Where is the "ignition module" ... was going to hit it with freeze spray IF the problem comes back. Also, based on pictures I found, is this the module that links to the key to the ignition?

Sounds right, you don’t need to buy anything to test the spark except one spark plug and couple of ground leads. But you need to run it enough so it dies again a home. If that test shows no spark, then buy a ignition module. Prolly should test all 4 coil locations or at least the wettest plug.