Ultra thin LiPos exist, a quick search found some as thin as 0.4mm. However I would imagine that packaging for product safety is a concern, as you’d also need to appropriately shield the cell from damage.
The thinnest commonly available coin cells are 1.2mm thick, which is better, but still will require a creative interpretation of "credit card size".
An ISO credit card is 0.76mm thick.
tldr; we're not there yet...
Protecting it would certainly be a challenge. If I were to try, my first attempt would be to sandwich it between two sheets of stainless. (0.76-0.4)/2 only leaves .18mm left over... that's 33 gauge. Maybe too thin to add enough rigidity? If you used 28 gauge stainless that would bring the sandwich to 1.04mm, which might be a better compromise.
It would certainly need some prototyping.
Apple gets 1 year of battery life from 225mah. So, it seems reasonable to get a month from 30mah. Not great, but hey, we're just entertaining the packaging requirement.