Together with @TarasBanakh 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. Finite affine and projective spaces are also well-known. Finite balanced Lobachevsky planes also exist (simplest example). If we need Lobachevsky plane where there exist no Pasch configuration, then we have unitals. 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.