Since Henry also discusses the property of being equationally Noetherian, I think the following observation is worth posting. And it is too long for a comment, so I post it as an answer.
The observation is that there exists a torsion free lacunary hyperbolic group that is not equationally Noetherian. In fact, such a group can be constructed directly. Alternatively, we can use the following theorem.
Theorem. Let $\mathcal H_n$ be the closure of the set of all $n$-generated torsion free non-cyclic hyperbolic groups in the space of marked group presentations. Then there exists a torsion free lacunary hyperbolic group $L$ such that the set of $n$-generator presentations of $L$ is dense in $\mathcal H_n$.
The proof uses the the same idea as in my paper mentioned by Henry. It is a bit too technical to be posted here (but it is just a page long and I did verify the details).
Now let us take the Ivanov-Storozhev non-hopfian group $G\in \mathcal H_2$. Since $G$ is not equationally Noetherian, there is a system of equations $S$ which is not equivalent to any finite subsystem on $G$. Let us assume that the coefficients in $S$ are written as words in generators, so we can consider the same system over $L$. Let $F$ be any finite subsystem of $S$.
Since $G$ is not equationally Noetherian, $S$ is not equivalent to $F$ over $G$. In particular, there is another finite subsystem $F_1$ of $S$ which contains $F$ and which is not equivalent to $F$ over $G$. Note that the property of a tuple of elements of $G$ to be a solution to a fixed finite system can be detected using some finite ball. Hence $F$ and $F_1$ are not equivalent over every group from some sufficiently small open neighborhood of $G$. In particular, $F$ is not equivalent to $F_1$ (and hence to $S$) over $L$. Thus we obtain
Corollary. The (torsion free lacunary hyperbolic) group $L$ is not equationally Noetherian.