How to achieve low power consumption with ATTINY85
What was missing was that the ADC needs to be disabled before powering it off:

ADCSRA &amp;= ~ bit(ADEN); // disable the ADC
power_all_disable();


This reduced the power consumption to ~16 μA. Since there are other components on the circuit, I don't expect to achieve the 1 μA advertised by the datasheet.