  A group $G$ is called mixed identity-free if for every $n \in \mathbb{N}$ and
every $w \in G \ast F_n$ there exists a homomorphism $\varphi: G \ast F_n
\rightarrow G$ such that $\varphi$ is the identity on $G$ and $\varphi(w)$ is
nontrivial. In this paper, we make a modification to the construction of
elementary amenable lacunary hyperbolic groups given by Ol'shanskii, Osin, and
Sapir to produce finitely generated elementary amenable groups which are mixed
identity-free. As a byproduct of this construction, we also obtain locally
finite $p$-groups which are mixed identity-free.
