P1/AMS series firmware releases are documented here:
P1 firmware updates require an internet connection. The most recent version is 01.06.01.02. Updates are not currently available as downloadable files. You should receive a notification on screen:
Turning the printer off and then restarting may force a check for updates.
The X1 printers also require a cloud connection for updates if they are currently on firmware 01.08.00 or older. Once updated to the most current release (01.08.02) future updates can be downloaded and installed offline from an SD card. Those updates will be available here:
Bambu announced in July that offline updates for P1 and A1 printers will be available in the future.