Hi, We develop wearable, such as smart rings.
Size 40x6x1 mm
Capacity – 30mAh (or suggest max. value, that you could provide)
Voltage: 3.7V, min: 3.0V, max: 4.2V
Output current:
Continues current 20mA
It has to fit inside the cylinder, which can be put on fingers.

I think your target size of 6mm width is impossible to make. The smallest we have ever made was 9mm. Working surface for the anode and cathode for such batteries isn't so big in pouch cells. Also remember that all those small batteries are made by hand and in small sizes it is hard to handle the foil.