P-Bass Tone pot Works As Volume/Tone Blend? NBD gone sideways.

It may just be the angle of the photo, but where is the tone cap?
Anyway, if this if factory new, have the manufacturer sort it out. No new bass should be accepted when defective.

 

If memory serves me correctly on a P-Bass there are two wires, black and white. The white wire goes to center lug on volume pot and the black goes to ground. In your pic it seems the white wire from pickup is going to left lug which isn't right. If this is an out of the box brand new bass I'd send it back if not comfortable checking it. If you are comfortable then change connections.

 

Hot (white) goes to one outer lug of the volume pot. That lug has a jumper over to the tone control's wiper (center) lug. The wiper on the volume control goes to the hot of the output jack. So the volume control should work correctly. All the way to one side the wiper is directly connected to the PUP hot and sending that signal to the output. At the other extreme of the volume pot the wiper is shorted to ground, shorting the output to ground also. The signal path to the tone control send the signal to the wiper so at one extreme there's 250K ohm resistance between the signal and the cap so it doesn't bleed much treble to ground. At the other extreme the signal is shorted to ground through the cap. That appears to be what you have (assuming the capacitor is hiding under the black wire in the photo, on the left side of the tone pot.

I'd do this first- unsolder both pickup wires from the pots and use some alligator clips to plug the PUP leads directly into an amp. Make sure both coils of the PUP are working correctly. If so, hook 'em up again and make sure the solder joins are sound mechanically and electrically. Reflow the solder on all the joints and retest. But that's because I used to manage a store and I'm good with rewiring and troubleshooting, and that assumes that I like the bass (not the model, but this specific chunk of wood and metal). If those don't apply to you and because it's a brand-new reputedly factory sealed box, I'd be calling the retailer who took my money and getting it replaced with a correctly functioning bass.

But then I don't see the point in getting a factory-sealed bass unless I'm paying the factory directly. Why didn't the retailer open the box, inspect the goods, and make sure everything was right?

 

If the two legs of the capacitor are touching, or if the leg that's soldered to the pot lug is touching the body of the pot, there's your problem. It seems to me this would be the most likely point of failure.

 

It appears the cap is edgewise under the bridge ground wire in the picture.

 

As @Killed_by_Death said, check where the capacitor is soldered to the potentiometer lug. Is it touching the pickguard?

Another quick test would be to disconnect the wire between where the pickup hot comes in and the tone pot. Does the volume now work properly?

- John