Converting my comment, i.e. construction of countably infinite, finitely-generated, amenable torsion group with exponential growth.

Pick any group $G$ which is countably infinite, finitely-generated, amenable and torsion, e.g. the Grigorchuk group. Then the (small) wreath product still has all the properties $\mathbb{Z}_2 \wr G$, since amenability and torsion are closed under group extensions.

The new group is of exponential growth, here's a direct embedding of a binary tree in a Cayley graph of $G$: Let $S$ be any generating set for $G$ and let $f$ be the generator that flips the bit at the origin. Pick using K\H{o}nig's lemma a one-sided injective path $p \in S^\omega$, i.e. such that all prefixes $p_0 p_1 \cdots p_{n-1} \in S^n$ of $p$ evaluate to distinct elements of $G$. Now obviously the elements $f^{w_0} p_0 f^{w_1} p_1 \cdots f^{w_{n-1}} p_{n-1}$ are distinct for distinct binary words $w$, so we have embedded a binary tree over the generators $S \cup fS$.

Converting my comment, i.e. construction of countably infinite, finitely-generated, amenable torsion group with exponential growth.

Pick any group $G$ which is countably infinite, finitely-generated, amenable and torsion, e.g. the Grigorchuk group. Then the (small) wreath product still has all the properties $\mathbb{Z}_2 \wr G$, since amenability and torsion are closed under group extensions.

The new group is of exponential growth, here's a direct embedding of a binary tree in a Cayley graph of $G$: Let $S$ be any generating set for $G$ and let $f$ be the generator that flips the bit at the origin. Pick using Kőnig's lemma a one-sided injective path $p \in S^\omega$, i.e. such that all prefixes $p_0 p_1 \cdots p_{n-1} \in S^n$ of $p$ evaluate to distinct elements of $G$. Now obviously the elements $f^{w_0} p_0 f^{w_1} p_1 \cdots f^{w_{n-1}} p_{n-1}$ are distinct for distinct binary words $w$, so we have embedded a binary tree over the generators $S \cup fS$.