Yes they do, for example: http://www.tannus.com

The trade off is that they're somewhat heavier and have more rolling resistance than a good air-filled tire.

I've found that keeping tires out of the sun makes the biggest difference. If kept out of the sun, bike tires can last a long time. One time I bought a bunch at the same time, and over the years I've regularly gotten one or two out of storage and found them satisfactory.

They had hard foam tire bikes when I was a kid but they didn't last long. And solid rubber tires are heavy. Decreasing rotating mass on a bicycle is always a good thing.