Can a battery produce more amperes as per Ohm's law?
You are confusing the actual current value flowing into a circuit and the maximum amperage rating of a (real, non-ideal) voltage source.

Any practical voltage generator has physical limits on its maximum current output. A battery is no exception. To know its limits you should read its technical specifications (the battery's datasheet, for example).

Therefore, a battery rated for 1A maximum, can provide any current below 1A, provided it is attached to a resistor of suitable value (according to Ohm's law).

Moreover, as @RogerRowland already pointed out in a comment, a battery, as any other real voltage source, has an internal resistance, so you should also take that into account. That means that if you connect an external load whose resistance is comparable to the internal resistance, you will experience a voltage drop at the terminals: the source won't provide its nominal voltage in this case, but a lower value.

As @WouterVanOijen already said in its answer, the Ah rating of a battery, technically called capacity of the battery, is a whole different story: it is an indication of the energy stored in the battery.

As a rule of thumb, you could use the capacity rating as a rough estimation of the maximum current the battery can provide (when fully charged): a battery with 500mAh capacity can provide 500mA for an hour continuously (or less, over a longer time span). This is called a 1C discharge rate (where 1C means "1 times the Capacity"). 

A fully charged battery can always provide currents that are below 1C (in the example 0.5C would mean 250 mA), but for higher current you may be at risk of overloading the battery.

Beware: as I said, this is is only a rule of thumb, though. Whether or not a battery will handle discharge rates greater than 1C must be checked for in its datahseet. 

See, for example, the datasheet of a Duracell coppertop AA cell.

Some excerpts:



In this case the internal resistance will be 120m&Omega; when fully charged (internal resistance increases as the battery discharges).

Keep in mind that the same capacity concept is a bit blurry, since the capacity rating depends on the load conditions. See for example the following graph from that datasheet, which assumes a load that draws a constant current regardless of the voltage applied (so it is not a simple resistor, but in this case calculations are easier, as you'll see):



Considering the rightmost curve, corresponding to a 100mA discharge current, you see that the number of "service hours" varies according to which cut-off voltage you choose, i.e. the voltage at which you consider your battery dead. 

If you choose your cut-off voltage to be 1.3V (red lines) your product will run for about 7h, so the cell sports a capacity of 7h&times;100mA=700mAh.

If, on the other hand, you choose your cut-off voltage to be 1.0V (green lines) your product will run for about 24h, so the cell sports a capacity of 24h&times;100mA=2400mAh.

You may also see that the leftmost curve corresponds to a 1000mA discharge rate and it is the maximum for which that cell is characterized. Although the datasheet doesn't explicitly say "max current=...", we can safely infer that that is the maximum current that the cell can deliver continuously without being overloaded. Note that I said continuously: a cell is able to provide higher current spikes for limited time (but in our example above nothing can be inferred from the datasheet about "pulse loads").

Note also that the case of a purely resistive load is covered by another graph:



The calculation of the capacity here is more difficult, since you would need to derive a graph of current vs time and perform an integral on it (i.e. the sum of the products &Delta;I&times;&Delta;t over the time interval for which the voltage is above cut-off). This difficulty comes from the fact that for a pure resistive load the current varies with time as the voltage slowly decreases due to discharging.

Finally, see that the discharge curves (and so the capacity) vary also with temperature. Here that datasheet provide only what happens for a 100mA discharge rate:



You can see that at 0°C the discharge is more rapid (less capacity): 100mA&times;16h=1600mAh@0°C vs 100mA&times;24h=2400mAh@21°C.
Therefore this kind of cell is better suited for operation at room temperature.