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We are interested in deformations of Baker domains by a pinching process in curves. In this paper we deform the Fatou function $F\left(z\right)=z+1+e^{-z}$, depending on the curves selected, to any map of the form $F_{p/q}\left(z\right)=z+e^{-z}+2\pi ip/q$, p/q a rational number. This process deforms a function with a doubly parabolic Baker domain into a function with an infinite number of doubly parabolic periodic Baker domains if p = 0, otherwise to a function with wandering domains. Finally, we show that certain attracting domains can be deformed by a pinching...