I found on Amazon a P3P battery made by Powerextra that sells for much less than the DJI batteries.  Does anyone have experience with these non-DJI batteries?

You won't even be able to start the motors
And if your drone crashes or sth similiar DJI won't take responsibility

I bought one of those for my P2V(+).  It has good reviews.  I've flown with it 15-20 times.  It seemed okay.  But the last time on Feb 4th it auto landed on me during RTH after just under 15 mins of flight.  The AC landed softly in a field.  I don't know what the percentage was for sure, but the voltage showed 10.6 volts which equates to just over 3.5 volts per cell.  I am not using it again until I can test it and I'm waiting for the wind to cooperate.  

Because of the nature of the smart battery and firmware issues I would never buy anything other than a DJI battery.

I've heard Horror stories that maybe if not at first on your first update the batteries will not be able to update in one no longer

if your P3 is out of warranty, and you are not worried about your drone dropping out of the sky, then give them a go.  I would personally advise against knock off batteries, but if you do buy them, then at least flight test them with props off, and when you start using them, put your warning at 50 percent, until you feel confident in the battery.

For your flight safety, it's not recommended to use non-DJI batteries.
And DJI Go app will recognize the battery automatically.