I read in a paper that all nilpotent uniform pro-$p$ groups of dimension $\leq 2$ are Abelian (Prime Decomposition and the Iwasawa $\mu$-Invariant by Hajir-Maire, Math. Proc. of the Camb. Phil. Soc. (2019) Section 5.1). This is probably obvious, but I am unable to prove this nor have I seen a reference.
This would mean $\mathbb{Z}_p \rtimes \mathbb{Z}_p$ is solvable but not nilpotent; what about $\mathbb{Z}_p^d \rtimes \mathbb{Z}_p$ when $d>1$? Here $\mathbb{Z}_p$ is the notation for the $p$-adic integers.
The group of $3\times 3$ upper triangular matrices with $1$ on the diagonal and entries in $\mathbf{Z}_p$ is 2-step nilpotent (and not abelian), and is a semidirect product $\mathbf{Z}_p^2\rtimes\mathbf{Z}_p$.
(The case $d=1$ is special: one can use the fact that every $2$-dimensional nilpotent Lie algebra is abelian.)