This is a very general question: are there known examples of Ramanujan behaviour of Cayley graphs obtained from family of finite p-groups?
${\mathrm{\bf Adjacency~matrix:}}$ Given a graph ${\mathcal{G}} = {\mathcal{G}}(V,E)$ ($V$=set of vertices, $E$=set of edges), the adjacency matrix $A({\mathcal{G}}) = (a_{ij})$ is given by the entries $a_{ij}$ = number of edges joining $i$-th vertex to $j$-th vertex.
If ${\mathcal{G}}$ is connected $k$-regular with $|V| = n$, the eigenvalues of $A({\mathcal{G}})$ are written as $k = \mu_0 \geq \mu_1 \geq \dotsc \geq \mu_{n-1} \geq -k$ ($-k$ is an eigenvalue depending on the parity of $n$). The eigenvalues $k$ and $-k$ are called trivial eigenvalues.
${\mathrm{\bf Ramanujan~behaviour:}}$ A finite, connected, $k$-regular graph ${\mathcal{G}}$ is called ${\mathrm{\bf Ramanujan~graph}}$ if any non-trivial eigenvalue $\mu$ satisfy $|\mu| \leq 2 {\sqrt{k-1}}$.
${\mathrm{\bf The~problem~is:}}$ for most of the infinite family of connected Cayley graphs, they are eventually non-Ramanujan. So, finding such infinite family, is often hard. The situation with finite simple Lie-type family with specified generating sets, is "almost" known.
The only example I know of is given by Alex Lubotzky long time ago on the Sylow-$p$-subgroups of $\operatorname{GL}(n,p)$ with the $n \geq 2$ (the family with $n$ varies).
I don't mind if it require the minimum number of generators bounded (or even fixed, say $2$). From the proof of coclass conjecture, perhaps one require the coclass also should diverge to infinity.
