Disclaimer: I haven't used much of Processing.js, so I am not the most knowledgeable...
Have you tried a simple loadImage() and image()? With luck, it can work if the browser supports it.
Otherwise, breaking down the Gif binary data is doable but is likely to be hard and slow.
You better use a desktop tool to decompose the animation in several images and to concatenate them in a PNG strip (faster to load than a bunch of separate images), then to display each part in succession.

Animated GIF's do not work natively in Processing.js. You will need the individual frames in separate images or use some form of spritesheet handling class function.

There will be a spritesheet handler class available for taking an image with several frames of an animation in it and return back an array of a PImage for you to iterate through in a sketch as you want soon.

I'm releasing a basic 2D game framework on the P.JS repo in a few weeks that has this spritemap handler class in it.