I will discuss an important "real-world" example of perfectoid spaces, the Lubin-Tate tower at infinite level. The etale cohomology of these perfectoid spaces agrees with some completed cohomology groups defined by Emerton, and can be used to realize the $p$-adic Local Langlands correspondence between certain $p$-adic representations of the groups $\mathrm{GL}_2(\mathbf{Q}_p)$ and $G_{\mathbf{Q}_p}$.