Let $A$ be the algebra of $\mathbb{R}\to\mathbb{R}$ locally Lipschitz functions. What is the vector space of derivations at $0$? The proof that for continuous functions there aren't really any doesn't seem to work in this case.
I was thinking about trying to define the tangent spaces for Lipschitz manifolds analogously to differentiable manifolds, but I couldn't work out even this simplest example. Can tangent spaces of Lipschitz, even topological manifolds be defined at all and how?
As Etienne pointed out in a comment, the book Lipschitz Algebras by Weaver is very much relevant. Section 4.7 introduces and describes derivations on the algebra of Lipschitz functions on a compact metric space. Weaver's constructions can be viewed as a way to introduce differentiable structures on metric spaces. See this paper by Gong.
I don't know of any concept of a tangent space to a topological manifold. But a Lipschitz manifold is a complete doubling metric space, and therefore one can define its tangent cones as (pointed) Gromov-Hausdorff limits of rescaled spaces $(X,\delta^{-1}d)$. Different sequences of scales $\delta_n$ can produce different limits, which means tangent cone is not unique in general. See section 8.7 of Nonsmooth Calculus by Heinonen.