A clay familiar can only be summoned during the Stealing Creationminigame from a sacred clay pouch (class 4). This class of clay familiar requires one piece of class 4 clay to create. As well as being able to fight, it's a Beast of Burden able to hold 18 items. It should be noted that it is impossible to dismiss or withdraw items from clay familiars while in fog. It gives access to the clay deposit special move, which deposits all carried items in your team's base.
When you die any familiar you had summoned will disappear and will drop any items they were holding on the ground where they were standing when you died. However unlike items carried by the player, items dropped this way will only be visible to the familiar's owner for a short period of time, similar to when items are dropped normally. Thus many players use familiars to hold items so that upon death the items may be reclaimed, preventing one from having to make new ones - this is particularly common with runes.
This article on a Summoning familiar has an associated dialogue page. Click here for a transcript of possible conversations with this familiar.