We study the Steiner Tree problem on the intersection graph of most natural families of geometric objects, e.g., disks, squares, polygons, etc. Given a set of $n$ objects in the plane and a subset $T$ of $t$ terminal objects, the task is to find a subset $S$ of $k$ objects such that the intersection graph of $S\cup T$ is connected. Given how typical parameterized problems behave on planar graphs and geometric intersection graphs, we would expect that exact algorithms with some form of subexponential dependence on the solution size or the number of terminals exist. Contrary to this expectation, we show that, assuming the Exponential-Time Hypothesis (ETH), there is no $2^{o(k+t)}\cdot n^{O(1)}$ time algorithm even for unit disks or unit squares, that is, there is no FPT algorithm subexponential in the size of the Steiner tree. However, subexponential dependence can appear in a different form: we show that Steiner Tree can be solved in time $n^{O(\sqrt{t})}$ for many natural classes of objects, including: Disks of arbitrary size. Axis-parallel squares of arbitrary size. Similarly-sized fat polygons. This in particular significantly improves and generalizes two recent results: (1) Steiner Tree on unit disks can be solved in time $n^{\Oh(\sqrt{k + t})}$ (Bhore, Carmi, Kolay, and Zehavi, Algorithmica 2023) and (2) Steiner Tree on planar graphs can be solved in time $n^{O(\sqrt{t})}$ (Marx, Pilipczuk, and Pilipczuk, FOCS 2018). We complement our algorithms with lower bounds that demonstrate that the class of objects cannot be significantly extended, even if we allow the running time to be $n^{o(k+t)/\log(k+t)}$.

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