AAAA cell’s are almost never used in flashlights, but are quite common in remote’s and doorbell’s.
AA and AAA are all over the world but casings take away a lot of the advantages in the smaller sizes, nominal sizes are:
AA = 14.5 x 50.5 mm (14500)
AAA = 10.5 x 44.5 mm (10440)
AAAA = 8.3 x 42.5 mm (none)

The AAAA light mentioned in the OP is the far right light in a review of well known member HKJ
https://lygte-info.dk/pic/Beamshot20110312-AAA/DSC_6500.jpg

The other lights are regular, mostly discontinued, AAA flashlights.
Maybe you get lucky and can find a D.Q.G Tiny AAA XP-E R2 or an Ultrafire UF-T1. Those are a bit thicker, but shorter.

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