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I couldn't find the caramelldansen base in this post, but here is a version that should work just as well.

Here is the original lick icon base.

To save these and have them be usable, you have to click the "free download" button underneath the image.

Some other good bases:

Dance Icon

Caipirinha

Run Run Run

You can find many other bases like these through DeviantArt.

How to make them into a GIF:

Step 1. Edit the base however you like (I used MSPaint)

Step 2: Crop and save each frame individually as PNGs (Note for the lick base: It's best to crop them to include the black boarder. This makes the GIF 50x50, which is the standard DeviantArt icon size and can easily scale larger if need be.)

Step 3: Go to ezgif.com/maker and upload all your frames

Step 4: Set "delay time" to 10, then click "Make a GIF!"

Step 5: Right-click the result and save it. Congrats! You used a GIF base!

edit THANKS @smellslikebot