Hi @Anonymous ,
Do you want to save multiple files into your local device once time within your app?
If you want to save multiple files into your local device once time within your app, I think the Attachments control could achieve your needs.
On your side, you could consider add a Edit form in your app, and add a SP List data source in your app, connect the Edit form to your SP List, then enable "Attachments" field data card in the Edit form, select the Attachments control within the form, then copy it (Ctrl + C), paste it in your other screen of your app. After that, you could use the Attachments control to upload files.
Please check and see if the following video resources could help in your scenario:
https://www.youtube.com/watch?v=kQVYid1p_vA
https://www.youtube.com/watch?v=UYK7yruBHDM
I have made a test on my side, please consider take a try with the following workaround:

Set the OnSelect property of the "Save" button to following:
ClearCollect(
FileCollection,
ForAll(RenameColumns(FileAttachments.Attachments,"Name","Name1","Value", "Value1"), {FileName: Name1, FileContent: Value1})
);
SaveData(FileCollection, "LocalData")
Set the OnStart property of the App to following:
LoadData(FileCollection, "LocalData", true)
Please consider take a try with above solution, check if the issue is solved.
Best regards,