Let's only consider central extensions (so we only miss a few cases when $p=2$). I denote by $C_p=\mathbf{Z}/p\mathbf{Z}$, to avoid confusion with $p$-adics.

So extensions are classified by $H^2(C_p^k,S^1)$. The commutator map yields a canonical homomorphism $\phi$ from $H^2(C_p^k,S^1)$ onto $\mathrm{Hom}(\Lambda^2C_p^k,S^1)$, and the latter is isomorphic to $C_p^{k(k-1)/2}$ (more canonically, the dual of $\Lambda^2C_p^k$). The kernel of this homomorphism consists of those 2-cocycle defining an abelian extension; since $S^1$ is an injective $\mathbf{Z}$-module the only abelian extension is split so $\phi$ is an isomorphism. Hence, the extensions are classified by $C_p^{k(k-1)/2}$, more precisely the space of alternating bilinear forms on $C_p^k$.

Next, the isomorphism class of the groups thus obtained are classified by the quotient of the latter by the action of $\mathrm{GL}_k(\mathbf{Z}/p\mathbf{Z})$. This quotient has exactly $1+\lfloor k/2\rfloor$ elements (number of types of alternating forms in dimension $k$, regardless of the field, including characteristic 2).

Note: For $k=1,2,3$ this makes $1$, $2$, $2$ isomorphism types of Lie groups. That for $k=2,3$ it does not match with Ian's answer (3, 8) is because he also considers quotients of order $p^k$ that are not $p$-elementary.

topologicalgroups. $\endgroup$ – YCor Apr 30 at 8:46