KöMaL 2023 A866

A graph is called 2-connected, if after deleting any point (and the edges adjacent to it) from the graph it remains still connected. Is it true that in any 2-connected graph with a countably infinite number of vertices it's always possible to find a trail that is infinite in one direction (i.e. a sequence of not necessarily distinct points $v_1, v_2,\ldots$ such that $v_i$ and $v_{i+1}$ is always connected with an edge, and all edges $(v_i,v_{i+1})$ are distinct from each other)?