No - you can't attach things to pages you can't see.
That's quite sensible though - users get really annoyed when you do things like that to them - they uploaded it, why on earth can't they see what they uploaded? You'll rapidly find them refusing to upload stuff and keeping local copies and then telling you that they aren't using confluence because they can't see the results.
But the original uploader still has the original attachment, and where I've seen this implemented, they immediately stop sharing it because they can't see it any more. It's human nature. And organisations that think they have a need for this are causing themselves problems.
I'm not aware of anything that would let you do this at all. Oddly enough though, Jira does it - it's possible to rig it so that someone can create an issue (with attachment!) and never be able to see it again (and then spend a lot of time whinging at the admins/helpdesk because they can't!)
A very dirty workaround:
You can create a mailbox and configure that mailbox in Confluence and request all users to send all their attachments as email-attachments to that mailbox.
And give permission only for those to access the space.
And I am wondering myself, why then a Confluence space is needed? May it is better to store things in Wiki, even if it comes from mail, so that anytime it can be opened up for everyone to use.
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