You can definitely add a flash banner to your site, but not in the banner position of a template that has a changeable banner. This is because the banner area does not support .swf files.
You can, however, choose a template that does not have a banner image area and place a Flash widget at the top of the page and into that your flash file. You can use any of the Responsive templates which includes the option to remove your banner area. Here's the steps for this:
1. In your Sitebuilder, go to Style > Change Template.
2. Click Responsive and select one of the available templates.
3. Then, go to Style > Designer.
4. Scroll down to the bottom and uncheck "Display Banner".
Once you've selected your template, here's how to add a flash file to your page:
1. Go to Widgets.
2. On the drop-down menu, click Media.
3. Drag and drop a Flash widget onto your page.
5. Once you have selected your file, it will display in the editing dialog box.
6. If necessary, edit the width, height, alignment and margin of the file.
7. Click Save.
If you upgrade to Yola Silver or Yola Gold, you will have access to the Site CSS feature and can use CSS coding to remove the banner area!