For an entire function f let N ( z ) = z - f ( z ) / f ′ ( z ) be the Newton function associated to f . Each zero ξ of f is an attractive fixed point of N and is contained in an invariant component of the Fatou set of the meromorphic function N in which the iterates of N converge to ξ . If f has an asymptotic representation f ( z ) ∼ exp ( - z n ) , n ∈ ℕ , in a sector | arg z | < ε , then there exists an invariant component of the Fatou set where the iterates of N tend to infinity. Such a component is called an invariant Baker domain. A question in the opposite direction was asked by A. Douady: if N has an invariant Baker domain, must 0 be an asymptotic value of f ? X. Buff and J. Rückert have shown that the answer is positive in many cases. Using results of Balašov and Hayman, it is shown that the answer is negative in general: there exists an entire function f , of any order between 1 2 and 1 , and without finite asymptotic values, for which the Newton function N has an invariant Baker domain.