Are there plastic materials that turn into ceramics?
The set of compounds you are looking for are known as Thermosetting Polymers. A few examples of these are:


Bakelite: This is used for making switches and electrical enclosures 
Melamine: This is commonly used for making food grade and microwaveable utensils and is an alternative to ceramic (at least in the food packaging industry)


You might also want to take a look at Thermoplastic Polymers, as they behave a bit like wet clay (they become soft on heating and harden on cooling and this can be done as many number of times as required)