Let F₀ and F be perfect subsets of the complex plane ℂ. Assume that F₀ ⊂ F and the set Ω = d e f F\ F₀ is open. We say that a continuous function f : F → ℂ is an analytic continuation of a function f₀ : F₀ → ℂ if f is analytic on Ω and f|F₀ = f₀. In the paper, it is proved that if F is bounded, then the commutator Lipschitz seminorm of the analytic continuation f coincides with the commutator Lipschitz seminorm of f₀. The same is true for unbounded F if some natural restrictions concerning the behavior of f at infinity are imposed.