The antenna is the limiting factor IIRC, the smallest module available is this thing: https://www.fdk.com/cyber-e/electronic_components/module/ble... which has the antenna as a slot in the package itself.
Six of them fit on the face of a dime, which is wild.
[1] The world's smallest Bluetooth® Low Energy Module, the new model "HY0021":
A trace at the circumference of the PCB, or a wire antenna that only take up space for the solder joint would do.
The battery/power supply alone is going to be at least an order of magnitude larger in volume.
Indeed, people do use rechargeable lithium batteries like the 50mAh CR322 (22mm by 3mm) or smaller CR311, but these would only be able to handle a 11mA EFR32BG29 output load for a few hours at most... Yet in a mostly dormant power save mode YMMV =3
Microcrystal can make these things to run at <50nA. https://www.microcrystal.com/en/products/real-time-clock-rtc...
All the BLE manufacturers are 1uA plus. It's infuriating.