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We construct a smooth compact n-dimensional manifold Y with one pointsingularity such that all its Lipschitz homotopy groups are trivial, butLipschitz mappings Lip(S^n,Y) are not dense in the Sobolev spaceW^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz(n-1)-connected, then Lipschitz mappings Lip(X,Y) are dense in N^{1,p}(X,Y)whenever the Nagata dimension of X is bounded by n and the space X supports thep-Poincare inequality.

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces