You can save files to the device's internal storage. Files saved to internal storage are containerized by default and cannot be accessed by other apps on the device. When the user uninstalls your app, these files are removed. The following code snippets would persistently store sensitive data to internal storage.
You should check the file mode to make sure that only the app can access the file. You can set this access with MODE_PRIVATE. Modes such as MODE_WORLD_READABLE (deprecated) and MODE_WORLD_WRITEABLE (deprecated) may pose a security risk.
Search for the class FileInputStream to find out which files are opened and read within the app.