I don't think you want to be extending ui-contact in this case, but rather using it as an internal element. I can't speak definitively because the code provided is too barebones to reveal intention, but I would imagine it would be something more like:
<template repeat="{{contact in contacts}}">
<ui-contact-chip contact="{{contact}}"></ui-contact-chip>
</template>
Where ui-contact-chip would be defined more like:
<polymer-element name="ui-contact-chip" attributes="contact">
<template>
<ui-contact id="contact" data="{{contact}}"></ui-contact>
<template bind="{{$.contact}}" if="{{$.contact.name}}">
<div>{{name}}</div>
</template>
</template>
</polymer-element>
This is just rough and spitballing, but the point is that extension should be thought of like class inheritance. You're saying that ui-contact-chip IS A ui-contact. What I think you want instead is to say that ui-contact-chip REFERENCES A ui-contact.