I found the following parts that should be what you need. Prior to replacing any parts however, try cleaning the internal pads for these buttons. Sometimes there is enough debris to prevent proper contact. In any case, these buttons have 2 components:
1) the underlying PC board that has the traces to be shorted by
2) the key pad which contains a conductive pad that shorts the two traces.
Included below are the 2 PC board parts as well as the shell components which will contain the push button assemblies.
https://www.ebay.com/itm/Official-PSP-1000-Direction-Directional-Pad-Button-Part-Replacement-Dpad-Flex/192249361904?hash=item2cc2f475f0%3Ag%3AJZUAAOSw95FZatsI&_sacat=0&_nkw=psp+parts&_from=R40&rt=nc&LH_TitleDesc=0%7C0
https://www.ebay.com/itm/IT012217024-PSP-1000-ABXY-CIRCUIT-BOARD-IT-PARTS-Replacement-Parts/162416242390?hash=item25d0c32ed6%3Ag%3Ad-8AAOSwdGFYuGx5&_sacat=0&_nkw=psp+parts&_from=R40&rt=nc&LH_TitleDesc=0%7C0
https://www.ebay.com/itm/PSP-1000-Full-Housing-Shell-Case-repair-parts-white-umd-door-button-sticker-New/141750370855?hash=item2100fb0227%3Ag%3AtyUAAOSw8GtZWd35&_sacat=0&_nkw=psp+parts&_from=R40&rt=nc&LH_TitleDesc=0%7C0
Dan