Setting a PHP Cookie within an AJAX Call [duplicate]

I’m pretty new to AJAX so I’m struggling a little with my functionality for my WordPress site.

I have a dropdown list of countries and when a user selects a different country, I would like to update the cookie (which is already set) and therefore the page should load the new cookie without reloading the page.

At the moment I have to hit refresh for this to show the correct country code from using the code below, whereas I’d have hoped AJAX would do this without refreshing the page:

<?php echo "Value is: " . $_COOKIE['country']; ?>

My code is below, but I’m sure some of it may not be needed or is wrong.

JS

$('.single-country-select').on("change", function (e) {
    var country_code_value = $('.single-country-select').val();
    var sendData = {
        action: 'country_change_ajax_function',
        country_code_value: country_code_value
    };

    $.ajax({
        url: ajaxadminurl.ajaxurl,
        type: 'POST',
        data: sendData,
        dataType: 'html',
        cache: false,
        success: function(data) {
            // SET COOKIE HERE??
        }
    });
});

functions.php

function country_change_ajax_function() {
    $country_code_value = $_POST['country_code_value'];
    setcookie('country', $country_code_value, time() + (86400 * 30), '/');
    exit;
}
add_action('wp_ajax_country_change_ajax_function', 'country_change_ajax_function');
add_action('wp_ajax_nopriv_country_change_ajax_function', 'country_change_ajax_function');

I’ve also got some code at the top of my header that sets the cookie automatically:

if(!isset($_COOKIE['country'])) {
    $get_user_country_code = simplexml_load_file("http://www.geoplugin.net/xml.gp?ip=".getRealIpAddr());
    if ($get_user_country_code === false) {
        $user_country_code = 'GB';
        setcookie('country', 'GB', time() + (86400 * 30), '/');
    } else {
        $user_country_code = $get_user_country_code->geoplugin_countryCode;
        setcookie('country', $user_country_code, time() + (86400 * 30), '/');
    }
}

Any help would be much appreciated. Thanks!