Many people, including many Members of Congress, behave as though there’s such a thing. One recent major example involved the 2012 Presidential campaign, when former U.S. Senator Harry Reid (D-NV) famously accused Mitt Romney, the Republican candidate, of not paying his (Romney’s) income taxes. Of course, this wasn’t true, and Reid knew it. But Reid also knew that he was safe from retribution, partly because the major mass media weren’t about to challenge him (and they didn’t) and partly due to the general belief that Article I, Section 6 protected him (it doesn’t). That Section simply says, “The Senators and Representatives…shall in all Cases, except Treason, Felony, and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and from the same, and for any Speech or Debate in either House, they shall not be questioned in any other Place.”
There IS immunity, but only from being arrested under certain specifically identified circumstances – and public slander isn’t one of them. A bit of background: This provision was put into the Constitution to protect Members of Congress from arrest by political enemies, and to foster free and open debate among the Members on issues of the day. As well, Great Britain not only still harbored notions of reclaiming the Colonies (that is, after all, what the War of 1812 was all about), but also was actively engaged in trying to do so.