As long as you are willing for the definition of "chewing cud" to be very broad, then caecotrophy may count.
Rabbit/hares excrete special pellets from the anus (different to regular faeces), which are reswallowed and redigested.
> Night feces (cecotropes) from this cecal fermented material are packaged in mucin and pass through the colon to the anus. These cecotropes are ingested by the rabbit and swallowed intact where they contribute substantially to the protein, mineral and vitamin nutriture of the animal.
* Harrison Pet \- not a particularly strong reference, but I don't think the science is in doubt here.
In the sense that the food is passed through the mouth twice for further digestion, it has some similarities to chewing the cud.