The "9 Ah" rating should be a measure of the useful power you can extract from it not some theoretical maximum.
The rating is usually based on a discharge time of 1 hour, i.e. if it is fully charged the battery will deliver 9A for 1 hour.
There is not likely to be any problem with currents smaller than that, so it will deliver 0.1A for 90 hours.
For very small currents, you also need to take account of the self-discharge rate of the battery (i.e. how fast it goes flat when it is not connected to a circuit) but that is only relevant if the expected battery life is months or years, not a few days as in your case.