Lenovo Thinkpad T450s: "Error: The non-volatile system UEFI storage is nearly full."

A customer of mine is using a Lenovo Thinkpad T450s since something like a year. The machine is running Debian jessie, with a keel out of jessie-backports, right now 4.5+73~bpo8+1. The lasted UEFI version is flashed / installed. The operating system is installed in "UEFI mode", with an extra EFI partition etc. Up until ~ four weeks ago, this setup was rock solid and stable: Lenovo Thinkpad and Debian, what could possibly go wrong?

Since four weeks, at each boot, the machine shows the error I've put into the subject. Here is an image of this:

Pressing Esc will continue the boot process, which worked "well"...for two more weeks. The message then changed from

Press Esc to continue or F1 to enter setup.

to

Clean up YES or NO

(sadly, I've got no image of this).

My client hit "YES", which erased the storage, as far as I see all of this currently, which rendered the machine unbootable. Afterwards, I've restored the "debian" boot entry, and the machine was happy again, booting up fine, etc. This lasted for some more days; since ~ one week, the message is popping up again at each boot.

I've tried to contact the Lenovo support four times on four different days, and gave up after spending ~ 30 minutes in the phone queue each time.

I've used all my $(your-favorite-search-engine-here)-skills these last days, and found next to nothing: Where this comes from, how to debug and, most importantly, how to fix this. As things stand currently, I guess, the machine will be unbootable again soon.

Any pointers highly appreciated!

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