<<Is there a way to restrict this for certain pages or whole spaces only?>>
Manleen's suggstion works at a page level. For whole spaces, go into Permissions for the space and choose whom can delete by checking or unchecking the Attachments-Delete permission.
And if you're watching the space or page, you can be notified of deleted attachments and if one is deleted, as long as you are using a version of Confluence that is 5.x (can't recall the point version) or greater, then you can restore it from Space Admin>Content Tools>Trash (in the pre-5.x days, restoring attachments was a quite difficult task.
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