An infinite-dimensional version of Gowers' $\mathrm{FIN}_{\pm k}$ theorem

We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of $c_0$ is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces developed by Todorcevic in order to obtain an Ellentuck-type theorem for the space of all infinite block sequences in $\mathrm{FIN}_{\pm k}$.