While the new example with a mixamo model works great, unfortunately the latest version of the FBX loader is messing up animated models exported from blender.
a fbx loader with the fbx loader as from 28/12/2017:

the same model loaded with the new fbx loader as from 06/01/2018:

I'm attached the cat model here:
cat exported from blender.fbx.zip
I've checked several other fbx exported from blender, they all suffer the same issue.
( tested using threejs 89 )
I'm investigating this now.
Actually your model was not displaying well even with older versions of the loader. Screenshot halfway through the animation:

@pikilipita fixed here: #13061
Thanks!
I've checked a dozen of animated models with this fix, works like a charm now. Awesome :)
Just a thing: when minyfing the file, I get this error:
SyntaxError: Unexpected token: punc (()
( line 2940 )
maybe you should change the line 2937 to:
parseSubNode: function ( name, node, subNode ) {
I've checked a dozen of animated models with this fix, works like a charm now
Sweet! Great to have a second set of eyes on this, there are only so many models I can test by myself.
maybe you should change the line 2937...
Oops, that's what it should have been already. I'll make a PR.