I’ve found these two discussions around the File Picker (I did search before asking!), and it seems clear that it’s not possible for now.
In that case, is there any intention on changing/supporting this sort of thing (File picker without closing the popup)?
To add: It looks like there was some more discussion in bugzilla.
Judging by the last activity, I’d say it’s safe to assume it’ll stay like this for a while.
I will say though (selfishly at that), it’d be great to have this, since it matches other browsers, which adds some familiarity for users jumping ship (to FF)