Display single most recent post in category on every page of category archive

I’m assuming that you have a loop to display the top (static) post and a second loop for the paginated posts:

[header]

[static latest post] loop 1

[paginated content] loop 2

[footer]

For loop 1 you may need to clearly identify the category in your custom loop.

$my_cat = get_query_var('cat');
$args = array(
    'posts_per_page' => 1,
    'cat'            => $my_cat,
    'orderby'        => 'date',
    'order'          => 'DESC',
    'post_not_in'    => get_option( 'sticky_posts' )
);