Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 5 / c / Solution

Yes. Quotient only the first coordinate modulo $\mathbb{Z}$. The resulting process lives on $\mathbb{T}\times {\mathbb{R}}^2$ and the image of $S_1$ is the radius-$ϵ$ ball about $(0,0,0)$. The two noncompact coordinates form planar Brownian motion, which is recurrent; during its infinitely many returns to a smaller disc, the independent circle coordinate has a fixed positive chance of lying in the required interval. The Strong Markov property then shows that the target ball is visited infinitely often. Lifting back proves that $S_1$ is hit at unbounded times.