Having a field permission were Nobody can edit the field if the field is anything other than blank
The field can be accidently overwritten but still needs to be able to be set
...
all users who want to protect the fields once they are set
Hey!
Unfortunately it is not possible to "conditionally" limit the edit permissions for a field.
What you can do is limit specifically what users can update, by right clicking on the field and selecting Field Permissions.
As a workaround, I guess you could have your team working form an interface. Within an interface you can play with visibility rules (see Rules on image below) to determine whether the field should be visible or not, and hide it if it is not empty (or other conditions). Even if hidden, you can show it elsewhere as a calculated (e.g. formula) field which is not editable.
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