A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time $\mathcal{O}^*(2^{\mathcal{O}(tw \log(tw))})$. Using their inspired Cut\&Count technique, they obtained $\mathcal{O}^*(α^{tw})$ time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of algorithms matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut\&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time $\mathcal{O}^*(3^d)$ and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth $d$. Similarly, we obtain time $\mathcal{O}^*(4^d)$ and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal.