The bases on this post are all JPEGs for some reason. That makes them hard to use properly. I will link to the originals and provide some additional ones, along with a tutorial on how to actually make them into a GIF.
To save these and have them be usable, you have to click the "free download" button underneath the image.
Some other good bases:
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 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!