A recently proposed ℓ = ∞ field theory model of tachyon dynamics for unstable bosonic D-branes has been shown to arise as the two-derivative truncation of (boundary)-string field theory. Using an ℓ∞ limit appropriate to stable kinks we obtain a model for the tachyon dynamics on unstable D-branes or D-brane anti-D-brane pairs of superstring theory. The tachyon potential is a positive definite even function of the tachyon, and at the stable global minima there is no on-shell dynamics. The kink solution mimics nicely the properties of stable D-branes: the spectrum of the kink consists of infinite levels starting at zero mass, with spacing double the value of the tachyon mass-squared. It is natural to expect that this model will arise in (boundary) superstring field theory.