Valid HTML in Template Part

Of course there is. Just do it like this:

$sticky = get_option( 'sticky_posts' );
$args = array(
    'post_type' => 'mission',
    'ignore_sticky_posts' => 1,
    'orderby' => 'date',
    'post__not_in' => $sticky,
    'posts_per_page' => $count
);

$latest_missions = new WP_Query( $args );
if ( $latest_missions->have_posts() ) {
    echo '<ul>';  // prints opening ul tag before list items
    while ( $latest_missions->have_posts() ) {
        $latest_missions->the_post();
        get_template_part( 'content-list', get_post_format() );
    }
    echo '</ul>';  // prints closing ul tag after list items

    wp_reset_postdata();
}