Together with [@TarasBanakh][1] we faced the problem described in the title. Let me start with definitions.

A *linear space* is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of subsets of $S$ where for any distinct points $x,y\in S$ there exists a unique line $L\in\mathcal L$ containing $x$ and $y$. We'll denote this line as $\overline{xy}$.

A *hull* of a set $A$ we will call a smallest set $\overline{A}$ such that $A \subset \overline{A} \land \forall x,y \in \overline{A} (x \not= y \rightarrow \overline{xy} \subset \overline{A}$). Having this hull definition we can easily define dimensional-based structures. In order to simplify writing, we will just write $\overline{a_1a_2...a_n}$ instead of $\overline{\{a_1, a_2, ..., a_n\}}$.

We'll call a set $A$ *collinear* if $\forall x, y \in A(x\not= y \rightarrow \overline{A} = \overline{xy})$. Set $P$ is called a *plane* if exist 3 distinct non-collinear points $x, y, z\in S:P=\overline{xyz}$. Similarly we can define set *planar* if hull of any non-collinear 3-subset of it is the same plane $P$. At last, we'll call a set $S$ *3-dimensional space* if there exist 4 distinct non-collinear and non-planar points $x,y,z,w \in S:S=\overline{xyzw}$.

We'll call 3-dimensional space *balanced* (due to *balanced incomplete block design* $BIBD$) when cardinality of all lines is equal and cardinality of all planes is equal as well.

And at last, we'll call a 3-dimensional space $S$ *Lobachevsky* if for every plane $P\subset S$, line $L \subset P$ and point $x \in P \setminus L$ there exist at least two distinct lines $\Lambda_1, \Lambda_2$ in $S$ such that $x\in \Lambda_i\subset P\setminus L$. At last we can formulate the problem.

>**Problem.** Does there exist *finite 3-dimensional balanced Lobachevsky* space $S$?

Some comments. Obviously there exist infinite any-dimensional [Lobachevsky space][3]. Finite [affine][4] and [projective spaces][5] are also well-known. Finite balanced Lobachevsky planes also [exist (simplest example)][6]. If we need Lobachevsky plane where there exist no [Pasch configuration][7], then we have [unitals][8]. So, we can see that there are infinitely-many balanced Lobachevsky planes. Still, I was not able to find any example of Lobachevsky 3-dimensional space.

  [1]: https://mathoverflow.net/users/61536/taras-banakh
  [2]: https://en.wikipedia.org/wiki/Steiner_system
  [3]: https://en.wikipedia.org/wiki/Hyperbolic_space
  [4]: https://en.wikipedia.org/wiki/Vector_space
  [5]: https://en.wikipedia.org/wiki/PG(3,2)
  [6]: https://math.stackexchange.com/a/104110
  [7]: https://mathworld.wolfram.com/PaschConfiguration.html
  [8]: https://mathoverflow.net/a/457894/515881
  [9]: https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/designs/bibd.html#sage.combinat.designs.bibd.BIBD_from_difference_family