The country code prefix for calling Christmas Island is +61 8 9164.
To make an international phone call, you should dial: "exit code of your country" + "country code of the country you are dialing to" + "the phone number you want to call".
For example: To call Christmas Island from the USA or Canada, dial: "011 +61 8 9164 + phone number"(011 is the Exit code of the USA)