Let X be the Cantor set of [0,1]. It is well-known that X is a perfect compact subset of [0,1]. By Tietze's theorem, C(X) = {f|_X ; f Î C([0,1]) }. Define d on D(d) = {f|_X ; f Î C¹([0,1]) } by d(f|_X) = f¢|_X.  d is a well-defined derivation (if f|_X = 0, then for each x₀ Î X there exists a sequence {x_n} in X-{x₀} converging to x₀. So f¢(x₀) = lim_n [( f(x_n) - f(x₀))/( x_n - x₀)] = 0, therefore f¢|_X = 0).
But d is not identically zero so by [Sak2, Proposition 3.2.1] d cannot be extended to a closed derivation in C(X). So that d isn't closable.
This example is due to O.Bratteli and D.W. Robinson ([cf. Sak2, P.59]).
Ref.
[Sak2] S. Sakai, Operator algebras in dynamical systems, Cambridge Univ. Press, 1991.