How do I output only country part of location field using the SimpleMap plugin?
Never used the plugin, but from the docs:

parts
This contains the locations address, broken down into its constituent parts. All values are optional so you'll need to have checks on any you use to make sure they exist.
A list of the available values can be found here.
To access the short version of any part, append _short to the end of its name. E.g. {{ myMapField.country_short }}.

Sounds like the country info is optional and may not exist on the model, hence the note to check if it exists or not first.
i.e.
{% if myMapField.country_short is defined %}
    {{ myMapField.country_short }}
{% endif %}