  We prove that every 3-connected 2-indivisible infinite planar graph has a
1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many
vertices leaves at most one infinite component, and a 2-walk is a spanning walk
using every vertex at most twice.) This improves a result of Timar, which
assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also
provide other results when the graph is bipartite or an infinite analog of a
triangulation: then the prism over the graph has a spanning 1-way infinite
path.
