How a user can be permitted to only one record of the same DocType.
@simigireesh, Set that record by using User Permissions to respective user.
Refer link - https://erpnext.org/docs/user/manual/en/setting-up/users-and-permissions/user-permissions
How can we allow user to only append but not edit a field.
For now you can allow the User to edit the field. Only appending permission is not available out if the box.
Please create a github issue for this explaining the use case.
Thanks