Is there a device/battery that can be charged while it's being used?
At the level of the actual battery cells, the answer is no:  "Charging" means current is flowing through the cells in one direction, and "discharging" means that the current is flowing in the opposite direction.  You can't have current flowing in both directions at the same time.

At a higher level, the answer probably is yes:  It is possible to build a power supply that


takes "bulk power" from an external source,
provides regulated power to a load,
uses the external source to charge its internal batteries and supply the load when the external source is available,
uses its internal batteries to supply the load when the external source is not available.


Some "USB power bank" devices will operate that way. Unfortunately, not all of them will do that, and they rarely advertise the feature on the package.  You just have to buy one and try it out.