Sources for spares:
Fairphone appears to still have (limited) stock of the following FP1 spare parts (at the time of writing - note that numbers behind the link are the number of clicks, and do not relate to stock):
This forum section (#market) is an unofficial channel to ask for and offer spares.
There may local communities near you that have a few (mainly used) spares. Note that these groups in general will not ship parts, but will happily meet in person to exchange parts and/or help repair. Check the #localcommunities tag and especially the #addressbook.
In addition to the communities above there are also #fairphoneangels that can help out in person and occasionally have spare parts.
A Swiss Fairphone partner also has some spares available:
FP1 secondhand spare parts on our stock
As mentioned above, for batteries there is a 3rd party replacement originally designed for another phone. There are forum reports that this battery can be successfully used in the FP1 with a spacer. With the obligatory warning that using this battery solution is at your own risk: 🇬🇧 🇩🇪 Generic battery to replace FP1 battery (⚠️see first post for warning).
There is no realistic expectation that new FP1 parts will be produced again. For discussion about not being able to produce new spares, see towards the end of this topic. The official discontinuation is discussed here.
A note about batteries: A lot of suspected battery problems (especially those where the battery suddenly doesn’t charge anymore) are not actually battery problems. In these cases the issue is often a build-up of dust in the usb port (if you’re lucky) or a broken usb port (less lucky). Try cleaning the USB port first. Batteries can also be charged externally with a universal battery charger (examples) and USB ports can sometimes be resoldered by more advanced repair shops (examples).
Also see the #batteryguide for more tips. A community discussion of alternative batteries that are capable of powering the FP1 is detailed here.