From either the Files App or the iOS Doc Picker in another app:
You can prevent the iOS Files app being able to access your FileBrowser connections by tapping the Config icon, then select: "Admin settings" / "Feature Switches" / and turn off "Allow Network Doc Picker".
This feature is available in FileBrowser and FileBrowserGO. See our comparison page to see which version fits your needs. Download a Free trial of FileBrowserGO.