As all the talk above indicates, there are a lot of variables here. But first off, pain isn't okay, or something to play through. Tired hands, some degree of aching from doing something new and difficult for extended periods, that's normal. But the kind of pain you're describing is not okay pain. All that "playing through it" will get you is carpal tunnel syndrome.
There are many potential variables, with neck width only one. Neck thickness, scale length, action (how high off the fretboard the strings are), how long of a reach to the lowest frets (which has to do with arm length compared with where the bass sits on your lap or hangs on a strap), the angle of the bass as you're playing it, how high or low you have the bass strapped, how sound your technique is (relative to your own body), how hard/fast you're playing, etc.
It can be a funny thing. My hands are on the small side for a bass player, so generally a narrower neck feels like a relief to me. But a couple years back I got a bass that has the biggest neck (for a 4-string) I ever played - A 1980 Yamaha BB800 - and I love it. The neck is fat, fatter than many Ps, but at the same time it feels great. But I can't play it quite the same way as I play my EBMM Sterling, my slimmest-necked bass. Generally, it's the same experience, but there are things that are just harder with the beefy neck, so I adjust.
I say, listen to the folks above suggesting different playing techniques. Unless you have long arms and big hands, one finger per fret on the first five frets is probably a bad idea. Using your pinkie and shifting rather than stretching are good ideas. The good kind of stretching - doing hand & finger stretches - could also help if done gently and patiently over time. There are lots of smaller men and women who play large-necked instruments, and the key is finding the ways to access the music you're trying to get out of the instrument in the way that's most comfortable.
Being able to one-finger-per-fret at the far end of the neck doesn't make you a good bassist. Playing the music you want to play does. Your path to that is yours - you own it. No one else does, no outside set of instructions does. There's a best way for you to fret notes, which has mostly to do with your body.
I agree with others that trying out a new neck is a valid idea. If you really love worn Ps and don't want to split parts, you might consider looking for what I have - I have a '75 P-bass with the A (Jazz width, 1.5" at the nut) neck option; it was made that way. Those seem to be less common but still turn up on the used market. I've seen a few over the years in the TB classifieds.
