Having the Country be a picklist value in the Chili Piper form

Requirement: Have Global Picklist for Country and State enabled within your SFDC org. Please confirm with your SFDC admin that you have this enabled if you are not one. 

You'll want to use the "CountryCode" instead of "Country" on the Lead object when mapping. 

Similarly for the Contact object mapping, you'll want to use either "MailingCountryCode" or "BillingCountryCode" instead of "MailingCountry" or "BillingCountry". 

Example:

Similar logic would apply as well if you are adding State and want the CP form to be a Picklist instead of a plain text input.