ATtiny85 not working on battery power

Probably some issue related to digispark USB bootloader (updating the bootloader would help in that case - some of the bootloaders used have done odd things, and a lot of progress has been made in making them smaller since the version that clone producers use was "frozen in time" years ago., I would also try pulsing reset low (with l jumper by hand and seeing if the pin that's connected tothe LED on the board, whichever one that is on those boards) flashes indicating bootloader running, or terribad breadboard that contacts jumpers but isn't making contact with pins of ICs.

You must have a 0.1uF decoupling capacitor. It is not optional and certainly not with the power supply onthe other end of a pair of cheap spindly jumper wires. , If you do not have any, buy some. You will need them frequently (most frequently used capacitor by far)

Rule out bad connections by checking voltage to the top of the pin on the chip (very useful test in general, the connectivity or voltage to the pin itself instesad of the soemthing supposedly connected to it.) and checking continuity with a digital multimeter. If you do not have a digital multimeter, buy one, they're cheap.