Yes -- and it's quite beautiful! The species is actually called Hevea brasiliensis, or Hevea, and looks something like maple. This wood has a high grain density and characteristics much like teak, which makes it a durable species for green flooring and for green furniture. An added bonus to using this wood for cutting boards would be its natural resistance to mold and fungus.
Rubber trees are farmed on plantations in Southeast Asia. The liquid latex, which is something like sap, is tapped from the trees over a lifespan of 25 to 30 years. Trees past their producing lifespan used to be burned, but now most are "reclaimed" and milled into lumber used for flooring and furniture. This makes Hevea a sustainable choice on two counts: less pollution from burning and a reclaimed source.
I have flooring samples of this beautiful, blond wood from Terra Mai, a company in California supplying FSC Certified Reclaimed Wood. You can see photographs of the plantations, the harvesting of latex, and the finished wood on Terra Mai's website.
You might want to read the information on the RubberwoodProducts.com website. I found this interesting and informative, but the description of the "Treated Rubber Woods" concerns me, especially if the wood is being used for cutting boards. This website describes treating the wood with preservative chemicals using a pressurized vacuum process. I'd check the source of the wood you are planning to use for the cutting boards, and make sure that it is not chemically treated -- or, at the very minimum, research what chemicals are used and what the possible effects might be.
You probably already know, but it does bear repeating -- FSC (Forest Stewardship Council) certification assures that every company in the entire chain of custody of a product operates in an environmentally responsible way. This "third-party" certification is something to look for and to support with your purchase -- even if it adds a bit to the cost.