The absolute max. rating of the output is -0.3V
I suppose that this means that there is a diode that clamps negative voltages.
What is maximum current this diode can accept?
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Yes. There are ESD protection diodes on all the pins. So taking any pin below the GND pin will turn on the pin to GND diode clamps.
The Abs Max specs refer to voltages and do not mention max currents. The output is mentioned, but the output is usually sized for the larger currents than the inputs.
We usually recommend 10mA or less for the current through the diodes. The less the better (for a lower forward drop).
If the negative voltage is expected as part of normal operation, then we recommend adding an external Shottky diode.