If you're using a code editor like Dreamweaver, do a "Save As" landing.php or something. From there log into the Dashboard, go to Themes and click Inspect. The new page type will show up.

Then you can go to your landing page, click Design and change the page type. You can then make your edits to that page type without affecting any other pages.

Thanks Kevin. What file do I start with so I can do the Save As? I'm going to be sounding a bit thick until I get going…

Here's how I would do it. Access your server via FTP and navigate to your theme folder. There should be files in your theme folder called 'full.php' or 'right_sidebar.php'. Download the file whose structure is the closest to how you want the new landing page to be. While it's on your local computer, change the name to something like 'landing.php' and upload it again to your theme folder again. Then visit Dashboard->Themes and 'Inspect' your theme and you should see your new 'landing.php' page show up. Now you can change the landing pages 'Design' to use your new 'Landing' page type.

Thanks, that's got me a bit further, but it's likely I'll be back…

I seem to have managed it! Thank you both