On a local QGIS installation you should have permission to create a
geopackage anywhere on the file system where you have permission to
write. (Could be different on a Citrix installation or something similar.)
What I noticed during teaching is that QGIS by default tries to save the
new geopackage in the QGIS installation directory. So if you just type
your new gpkg name without using the "browse" button, you get the error
message for not having permissions to create the gpkg.
So, just try to pick a directory where you can create new files.
@developers: Is this behaviour on purpose, or could/should it be changed
to point to the user's home directory by default?