= Solution
Take $G=\mathbb C^*$ and the <complete analytic function> generated by a logarithm germ. Its continuation surface identifies with $\mathbb C$ via $w\mapsto[\log]_z$ with $z=e^w$ and logarithm value $w$. Thus $\pi(w)=e^w$. It is a <universal cover>; the deck transformations $w\mapsto w+2\pi in$ act transitively on each fibre, so it is also a <regular covering map> in the deck-transitive sense. \b[It is not bijective], since every nonzero $z$ has infinitely many logarithm values. It also meets the alternative convention in which “regular” merely requires an evenly covered neighbourhood at each base point.
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