How to use WordPress HTTP API to download file from remote location

Question

So this PHP code works for me:

    $ch = curl_init( TCS_CPDF_REMOTE_ZIP );
    curl_setopt( $ch, CURLOPT_RETURNTRANSFER, true );
    $data = curl_exec( $ch );
    curl_close( $ch );
    file_put_contents( TCS_CPDF_LOCAL_ZIP, $data );

but when trying to use the WordPress HTTP API:

    $the_body = wp_remote_retrieve_body( wp_remote_get( TCS_CPDF_REMOTE_ZIP ) );

    file_put_contents( TCS_CPDF_LOCAL_ZIP, $the_body );

I end up getting a 0KB file, so the above WordPress version is not working.

So how do you download a file from a remote location using the WordPress API?