We report the first observation of single-shot soft x-ray laser induced desorption occurring below the ablation threshold in a thin layer of poly (methyl methacrylate)--PMMA. Irradiated by the focused beam from the Free-electron LASer in Hamburg (FLASH) at 21.7 nm, the samples have been investigated by atomic-force microscope (AFM) enabling the visualization of mild surface modifications caused by the desorption. A model describing non-thermal desorption and ablation has been developed and used to analyze single-shot imprints in PMMA. An intermediate regime of materials removal has been found, confirming model predictions. We also report below-threshold multiple-shot desorption of PMMA induced by high-order harmonics (HOH) at 32 nm. Short-time exposure imprints provide sufficient information about transverse beam profile in HOH's tight focus whereas long-time exposed PMMA exhibits radiation-initiated surface ardening making the beam profile measurement infeasible.