Existence of formal lines over fields of positive characteristic

Ascertain whether formal lines, as defined by a k-algebraifold L in standard form with D_L free of rank one and some derivation \partial \in D_L surjective as a map L \to L, exist over fields k of positive characteristic; either construct such an example or prove nonexistence.

Background

Formal lines are the algebraifold analogue of lines used to define curves and geodesics algebraically. Over characteristic zero (e.g., k[t] with the usual derivative), formal lines exist.

For fields of positive characteristic, k[t] fails to be a formal line due to the behavior of derivatives of \ell-th powers, and the authors question whether any formal line exists in positive characteristic.