Yes, it is possible turn off iCloud backup and synchronization anytime and retain a local copy of your data.
First ensure that Pillow has finished synchronizing your data:
Open Pillow on your iPhone.
Tap on the "Settings" icon in the top right corner.
Scroll to the "Data & Services" section.
Tap on "Synchronize now" and wait until the synchronization has finished successfully.
Then you can turn off the "Sync with iCloud" switch in order to disable this feature. You can optionally turn off iCloud access for Pillow from the "Settings" app:
Open the "Settings" app on your iPhone.
Tap on your name at the top.
Tap on "iCloud".
Under the section "Apps using iCloud", select "Show all".
Turn the switch for Pillow to OFF.