Yes.
The golden-fronted woodpecker (Colaptes mexicanus) inhabits semi-desert and desert regions. In South America, the campo flicker (Colaptes campestris) is found, in particular, on the pampas, where there are no trees. When conditions are suitable, it prefers to nest in cavities, and in treeless areas, it digs burrows in cliffs or old termite mounds.
The ground woodpecker (Geocolaptes olivaceus, in the photo) from South Africa inhabits open spaces, forages on the ground, and builds nests in burrows.
There are also known cases where 'typical' woodpecker species nest in treeless areas — for example, the great spotted woodpecker has been observed in the Orenburg steppe.
