Let X be a Berkovich space over a valued field. We prove that every finite group is a Galois group over (B)(T), where (B) is the field of meromorphic functions over a part B of X satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over K(T), where K is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over Z and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite

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... roup is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over Z.