We present a new class of nanoscale experiments on cuprate superconductors in the current-carrying state. These experiments are aimed at studying how the high-T<sub>c</sub> order parameter responds, in both real and reciprocal space, when driven by a sizable phase gradient. First, scanning tunneling spectroscopy was performed on current-carrying YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6+x</sub> (YBCO) thin-film strips, to reveal a remarkable suppression of the low-energy Andreev states indicating an increase in the local phase fluctuations. Second, transport measurements were made on optimally-doped YBCO nanostructures, to show anomalous current-voltage nonlinearities consistent with the formation of current-driven phase slip lines. These results are discussed in the general context of non-rigidity of the high-T<sub>c</sub> order parameter under electrodynamic perturbation.