The chip's maximum operating voltage is 5.5v with an absolute maximum of 6v
If you do decide to use a 9v battery, then think about how much energy is actually stored inside such a battery... typically you're looking at around 400-500 mAh ... and they're relatively weak at pumping out current, they're designed more for alarm clocks, radios etc.
If you have two 1w speakers running them loudly and continuously, that means the amp board will consume about 2.2 watts (the datasheet says around 83% efficiency for the chip) so 2.2w / 9v = 0.25 A of current ... 9v batteries aren't really designed to give 0.25A continuously, the voltage will drop quite quickly.
So even if you're using a dc-dc converter to convert the 9v to let's say 5.5v with maybe 90% efficiency, you're probably going to get around 600-700 mAh of battery life.
Think about how much power you're gonna send to speakers, how loud will those things be... what speakers are you gonna use? If you're only gonna use 0.5w or 1w speakers, then you can live with smaller voltage, check datasheet :
THD+N = 1%, f = 1KHz, RL = 4Ω: 0.85w @ 3.2v , 1.3w @ 3.6v 2.5w @ 5v
So if you're no planning on using more than 1w or so, then you could simply use 3 AAA or AA batteries and have more actual life ... 3 rechargeable batteries give you ~ 3 x 1.1v = 3.3v ... 3 x 1.25v = 3.75v and 4 rechargeable batteries give you up to 5v
Rechargeable AAA batteries give you around 800mAh so 4 AAA batteries would give you the same amount of energy as a 9v battery without having to use a dc-dc converter.
AA batteries would give you around 1800-2500mAh so would last much longer.
You may want to keep the output power to under 1w anyway, look at the THD graphs in the datasheet, see page 5 :
https://www.diodes.com/assets/Datasheets/PAM8403.pdfWith 3.3v input, you can see the distortions jump as soon as you go over 1w and with 5v the distortions increase as you go over 2w