whatever9:
I have the chip plugged into the socket on the Tiny AVR programmer that is connected to a USB port.
So the answer to my question is "yes".
When I measure the voltage on other pins that have pull ups activated, the voltage is about 5 V, so I don't know why PB0 is acting differently.
Because the Tiny AVR Programmer includes an LED and series resistor connected to pin zero (MOSI).