Custom page template with entirely different design. Is it possible in?

Short answer is, yes it’s possible.

That is the very purpose of having a template hierarchy. Different templates having different functionality and design based on your needs.

For example, you can create a tag-facebook.php and make it look like Facebook, while having a tag-twitter.php that looks nothing like Facebook, but just like Twitter!

There is one thing you should consider. Most of the page templates have the following structure:

// Header
<?php get_header();?>

// The page's content is here

// Footer
<?php get_footer();?>

If you follow this structure, you can’t fully customize your page since get_header() and get_footer() will always include the same header and footer templates in your page.

Unless, you define different headers and footers to use in different designs.