Do I need to decouple an LM386 if it is run from a battery?
And more to the point here, the 50dB PSRR of LM386 is given with a 10uF bypass cap:



Without it... you are in Murphy's Law territory.

And I can tell you from teardowns of some Japanese (Uniden) cordless phones, which use the JRC version of that chip (for the internal [typically] 1/4W speaker), that even they don't omit the 10uF cap for that chip, and phones aren't known for best room audio quality. And they also have some 1000uF board-entry decoupling before that, so it's not like they run this IC straight from the wallwart. The 10uF cap is placed at about 0.5cm from the 386 on those. They did cut the coupling output cap from the 220uF (JRC datasheet recommendation) to 47uF though (which I assume is good enough for voice high-pass), so they were not exactly throwing their money away with no reason. These phones used to be made one every 3.2 seconds at one point, if Wikipedia is to be believed... so optimizing parts cost surely was on their mind.