How do I create a .tpl file for a panel pane? So, I want to add some HTML to surround the output of a Panel pane. I know the existing pane is rendered with:

panels-pane--entity-field-extra.tpl.php

If I edit this file, of course, every pane using it will show the changes.

If I create a new .tpl, say panels-pane--entity-field-top.tpl.php how do I apply it to a particular pane?
You can refer Documentation on Panels Pane template, which provides details on how to define a template for Panels Pane. If you want to add your own template name you can also do that by implementing template_preprocess_panels_pane in your theme or module file. Example for same is also available on documentation page.