ESD protection for battery powered device
Looking at the audio amplifier absolute maximum ratings, Vin is limited to the supplies at Vcc + 0.3V and -0.3V

This usually implies a small schottky is embedded within the device (but we cannot be certain and besides they can't handle much - perhaps a couple of mA). I would use schottky clamps (to Vdd and ground) on the audio inputs (at the input pins after the input resistors).

Note that the inputs are very high impedance (they do not appear in the filter equations implying that the input resistance is so high that it does not affect the input filter performance).

The current won't be huge, so choose devices that have &lt;0.3V forward voltage at perhaps 20mA. If an event does occur the actual pins won't exceed their maximum ratings.

Look at a clamp diode as ensuring that the input pins cannot exceed Vdd + diode drop or go lower than ground - diode drop.

The comparator has a high impedance output and would be quite susceptible to ESD hits; is the pullup on your board? If so, I would again clamp the output to both the pullup voltage and ground using schottky devices unless the threat is quite high, where I would use devices designed for the task. (See link below).

Schottky clamps in this type of application are quite effective against ESD.

If the pullup resistor is not on your board, I would probably use a zener as they tend to be a bit more precise than other solutions - TVS device clamp voltages have a wide range from engaging to full clamp currents.

If you want to use a clamping array designed for ESD protection, there are a number of solutions.

I won't go into the pros and cons of the different devices here but ESD issues are really no different across most devices, battery powered or not.

Decent write-up at Murata.