Following our previous work on metamaterial high-impedance surfaces made of Hilbert curve inclusions, here we theoretically explore the performance of the high-impedance surfaces made of another form of space-filling curve known as the Peano curve. This metamaterial surface, formed by a two-dimensional periodic arrangement of Peano curve inclusions, is located above a conducting ground plane and is shown to exhibit a high surface impedance at certain specific frequencies. Our numerical study reveals the effect of the iteration order of the Peano curve, the surface height above the conducting ground plane, and the separation distance between adjacent inclusions.