Example of non-amenable group which is the inverse limit of amenable groups

Question

1) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$?

2) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$ such that the kernels $\ker \varphi_i$ of projections $\varphi_i \colon G \to G_i$ are compact?

Answer 1

The paper "Amenable semigroups" of Day answers the first question. The free group is an inverse limit of amenable groups.

The second question remains open.