With all PORTB pins LOW, I still read ~1V on all pins.
New development : while I can't find a short, I've just found out something interesting.
When the PIC is in the socket, there is slight continuity between MCLR and all port pins when the negative test lead is on MCLR, not the other way around. I tried it directly on the PIC, out of the socket, and it doesn't show any continuity at all. Testing the empty socket doesn't show continuity either.
Again, the same pic powered outside the board works fine, so the board is the problem.
There are no shorts betweem pins, neither between Vdd and any pin as far as my DMM can tell...
I am clueless.