You can insert an animated GIF image as a background image or banner image.
This image was made using the animation feature in Google Photos.
This involves using the animation feature in Google Photos. The procedure is explained by Peggy K, a top contributor in the Blogger forum. You can find her post about this in Google+. She explains about using it in Blogger but as you can see it also works in Google Sites. You should view her detailed instructions but here is my condensed version:
- In Google Photos click the + icon at the top (on mobile click the 3-dot icon)
- Select Animation
- Click to select each image you want to include
- Click the Create button
- Click the more actions icon (3 dots) and add the animation to a shared album
- Right-click the animation and copy the image URL
- In your site click Insert, select Image and then Web address (URL)
- Insert the URL you copied in step 6
To make a slower animation you can use an editor like LunaPic. You can upload your fast GIF, change the speed, save the image and upload it to Google Photos, get its URL and insert it into your page. (Don't forget to move it into a shared album)