Select field - Let the user add new custom value

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Sounds promising.

For the time being - I handled my requirements with a different approach.

Note: I didn't check if that could be possible with hooks or other customisations over the select field.
Maybe there is a workaround with that approach.

But here is my solution.
Created a Settings Page for that post type.
Created a field for the settings page. It will accept the values (or the combination values - labels) that we want for the select field. Set the field as clone-able. Create some sample entries for the field.

Then for the original select field that we want those values - use the callback: callable_function for the options.
In functions.php - create the callable_function - make it get the values from the settings page - prepare the options and return them for the select field.

Later on, if the user needs to add more values (not on the fly though while editing/creating the post) - he can go and add more values from within the settings page.