I'm not sure that the OpAmp inputs can go as low as 6mV. You can test this by increasing the voltage to e.g. 7mV and see what happens. Also note that the offset voltage, according to the data sheet, can be as high as 2mV in low power configuration, which is a third of your input signal...
Thank you for the answer...
... And please disregard my ignorance, but to properly understand all factors, where does opamp offset voltage affect the output - before or after amplification? (i.e. is it [ input + offset ] * gain or input * gain + offset?)
Offset is measured at the input, so an offset of 2mv would be either added or substracted from the input signal. If would explain up to 300mV difference in the output signal. Try to use "high power" mode, there the offset is at most 1mV.
Thank you for the explanations. It looks like I will have to use a chopper amplifier with this, since the max of the input signal strength will be rarely reached; most often it will be around 3-4 mV, so I don't think that any on-chip opamp will be able to cut it...