Do connected, locally euclidean, Hausdorff, topological groups exist wihich are not second-countable?

Question

It is well-known that there exist connected, locally euclidean, Hausdorff, topological spaces which are not second-countable, i.e. the topology has no countable base. One example is the 2-dimensional Prüfer-manifold, see e.g. R.Nevanlinna: Uniformisierung (in German). If we only replace topological space by topological group, does there still exist an example which is not second-countable? I do not see that the Prüfer-manifold could be endowed with a group structure which makes it a topological group, but someone might come up with a totally different construction. Or someone can prove that such a topological group does not exist? Or this is an already known result?