WordPress wp_link_pages show only next/prev link

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I’m not sure why it’s not working.
Did you try with the defaults but only changing the “page” and clearing the before/after?

wp_link_pages( array(
    'before'           => '',
    'after'            => '',
    'link_before'      => '',
    'link_after'       => '',
    'next_or_number'   => 'next',
    'separator'        => ' ',
    'nextpagelink'     => 'Next page',
    'previouspagelink' => 'Previous page',
    'pagelink'         => '%',
    'echo'             => 1
    )
);

Alternatively you could always “hard code” the links to the single.php.

I mean that doesn’t really fix you issue, but you know..

<?php global $page, $numpages, $multipage, $more;
if ( $multipage ) {
    if ( $more ) {
        $prev = $page - 1;
        if ( $prev ) {
            $link = _wp_link_page($prev).'Previous page</a>';
            $output .= apply_filters( 'wp_link_pages_link', $link, $prev );
        }
        $next = $page + 1;
        if ( $next <= $numpages ) {
            if ( $prev ) {
                $output .= ' ';
            }
            $link = _wp_link_page($next).'Next page</a>';
            $output .= apply_filters( 'wp_link_pages_link', $link, $next );
        }
    }
echo apply_filters('wp_link_pages', $output);
} ?>

Modified straight from post-template.php/wp_link_pages()