Get last post in category thumbnail

Try this:

<?php 
    $the_query = new WP_Query( 'showposts=1&cat=55' );
    if($the_query->have_posts()) : 
        while ($the_query -> have_posts()) : $the_query -> the_post();?>
          <a href="https://wordpress.stackexchange.com/questions/144864/<?php the_permalink(); ?>"><?php the_post_thumbnail();?></a>
 <?php endwhile;
    endif;
    /* Restore original Post Data */
    wp_reset_postdata();?>

Replace cat=55 with the category ID you want and/or replace showposts=1 with the number of latest posts you want to show if you want more than 1.

If you want a different size for your thumbnail, you can use one of the followings (taken from http://codex.wordpress.org/Function_Reference/the_post_thumbnail):

the_post_thumbnail();                  // without parameter -> 'post-thumbnail'
the_post_thumbnail('thumbnail');       // Thumbnail (default 150px x 150px max)
the_post_thumbnail('medium');          // Medium resolution (default 300px x 300px max)
the_post_thumbnail('large');           // Large resolution (default 640px x 640px max)
the_post_thumbnail('full');            // Full resolution (original size uploaded)
the_post_thumbnail( array(100,100) );  // Other resolutions