Q:

Example of $3$-dimensional $p$-adic group

Example of $3$-dimensional $p$-adic group:
I am searching some examples of $3$- dimensional $p$-adic groups.
However I have touched one such group but I am not sure.
Consider the group $\{\pi \in SL_2(\mathbb{Z}_p) \ | \  \pi \equiv \begin{pmatrix} 1 &0 \\ 0 &1 \end{pmatrix} \mod p \}$. 
As far as I collected information, there is a $3-$ dimensional Lie algebra associated to this group and hence it is $3-$ dimensional p-adic group.
Can someone share some more details about this group?

A:

For $p$ odd prime 

Show $$\log : I+ p^2 M_2(\Bbb{Z}_p) \to p^2 M_2(\Bbb{Z}_p),\qquad \log(A) = -\sum_{k=1}^\infty \frac{ (I-A)^k}{k}$$ is injective (from $\log(I+p^k M) = p^k M+O(p^{2k})$)
Show $$\exp : p^2 M_2(\Bbb{Z}_p)\to I+ p^2 M_2(\Bbb{Z}_p) $$ is injective (from $\exp(p^k M) =I+ p^k M+O(p^{2k})$)
$\exp \circ \log = Id$ from our knowledge of complex analysis which extends to formal series thus to p-adic series.

Thus the Lie-algebra of $I+ p^2 M_2(\Bbb{Z}_p)$ is a $4$-dimensional $\Bbb{Z}_p$-module.
Then substract $\log(\det(A)^{-1/2} I)$ to find the image of the restriction to $I+ p^2 M_2(\Bbb{Z}_p) \cap SL_2(\Bbb{Z}_p)$.
The difference with real Lie groups is that $\exp$ doesn't have to extend to the  $\Bbb{Q}_p$-vector space generated by $Lie(G)$.

