  We show that there are CNF formulas which can be refuted in resolution in
both small space and small width, but for which any small-width proof must have
space exceeding by far the linear worst-case upper bound. This significantly
strengthens the space-width trade-offs in [Ben-Sasson '09]}, and provides one
more example of trade-offs in the "supercritical" regime above worst case
recently identified by [Razborov '16]. We obtain our results by using
Razborov's new hardness condensation technique and combining it with the space
lower bounds in [Ben-Sasson and Nordstrom '08].
