There are at least three role-playing games set in Flatland. From most to least developed:

  1. The Original Flatland Role Playing Game: Very detailed and nicely produced re-envisioning of Flatland as a campaign setting, including Planiverse-style physical explanations of how, for example, Flatlanders can hold objects (suction might be enough, because airtight seals are easy to form in 2D). Almost half the 97-page book is taken up by a reprint of the original Flatland, but it's hard to begrudge the author his $6.99, because all of his profits go to Doctors Without Borders. Note that the free Forgotten Futures Compendium contains a Flatland adventure by the same author, "The League of Extraordinary Geometers". Check out the sampler.
  2. Edward Abbot Abbot's Flatland (Inflated) the RPG: Somewhat less detail-oriented rendition of Flatland as an RPG, as you might guess from their mistaking the name of Edwin Abbott Abbott in the title. There are two expansions, an adventure called The Rescue of the Polynomial Princess and a Flatland monster manual called More Things to Kill. Compatible with The Original Flatland Role Playing Game thanks to conversion tables in the former.
  3. Fudge Flatland: A short sketch of how to create Flatland adventures with the Fudge roleplaying game, which is itself a very simple system. Freely available on the Web.