For continuous γ, g:[0,1]→(0,∞), consider the degenerate stochastic differential equation dXt=[1−|Xt|2]1/2γ(|Xt|) dBt−g(|Xt|)Xt dt in the closed unit ball of ℝn. We introduce a new idea to show pathwise uniqueness holds when γ and g are Lipschitz and $\frac{g(1)}{\gamma^{2}(1)}>\sqrt{2}-1$ . When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131–149] with $\gamma=\sqrt{2}$ and g≡c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C3 boundary and suppose $h\dvtx \widebar D\to \mathbb {R}$ Lipschitz on $\widebar D$ , as well as C2 on a neighborhood of ∂D with Lipschitz second partials there. Also assume h>0 on D, h=0 on ∂D and |∇h|>0 on ∂D. An example of such a function is h(x)=d(x,∂D). We give conditions which ensure pathwise uniqueness holds for dXt=h(Xt)1/2σ(Xt) dBt+b(Xt) dt in $\widebar D$ .