A plane continuum is constructed which has span zero but is not chainable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3, author = {L. C. Hoehn}, title = {A non-chainable plane continuum with span zero}, journal = {Fundamenta Mathematicae}, volume = {215}, year = {2011}, pages = {149-174}, zbl = {1238.54013}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3} }
L. C. Hoehn. A non-chainable plane continuum with span zero. Fundamenta Mathematicae, Tome 215 (2011) pp. 149-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm211-2-3/