A proper arithmetic surface with infinite étale fundamental group

We give an example of a proper, regular, integral arithmetic surface over $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}$ endowed with a section, and whose étale fundamental group admits an infinite pro-$2$ quotient. The construction is a geometric analog of the Golod–Shafarevich towers for rings of integers of number fields.