Making a light bass is something that takes attention to ALL the parts. Here's a Warmoth build that came in at 7.31 pounds (aka 3314 grams on my postal scale):
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Note that it's a 5 string - a similar build I did for a 4 landed it at 6.6 pounds. Anyway, this body was from their showcase - I knew the weight on it (3 pounds, 4 ounces) before I bought it. It's roasted swamp ash, the neck is roasted maple (with graphite reinforcing bars, not steel).
The bridge is a Hipshot A in aluminum, the tuners are Hipshot Ultralites, the neck mounting plate is aluminum; even the knobs on the thing are aluminum (plastic is even lighter). The pickup on this is heavier than a P pickup - you'd save about 80 grams if you used a standard P pickup.
Anyway, the lesson here is it's not just about the body and neck - you have to get light parts everywhere if you want a light bass. And whatever you do, on a bass this light, DO NOT use standard Fender style clunky steel tuners - half a pound of extra weight out on the end of your neck is something you really don't want.
One final note: if you want the lightest body possible, roasted swamp ash is your friend - call Warmoth and order it over the phone. Ask for their lightest blank - they will charge you something like $40 to do that (they won't do that over the web site - it's a kind of off menu option), but you will get a light body that way - I've done that a few times, and even my heaviest 5 string (2 pickups) comes in at 7.75 pounds, and they all (because of the light tuners) balance perfectly.
Oh yeah - Warmoth makes a G4 body that's got skinnier horns than a P bass - you can get the P pickup and all that, but it'll be a bit lighter than a strict P shaped body - it looks kinda like a G&L SB-1 body, so it's "Fender-esque", but a more modern shape than the old P shape.