The application of X-ray photon correlation spectroscopy (XPCS) at free-electron laser (FEL) facilities enables, for the first time, the study of dynamics on a (sub-)nanometre scale in an unreached time range between femtoseconds and seconds. For soft-matter materials, radiation damage is a major limitation when going beyond single-shot applications. Here, an XPCS study is presented at a hard X-ray FEL on radiation-sensitive polymeric poly(N-isopropylacrylamide) (PNIPAM) nanoparticles. The dynamics of aqueous suspensions of densely packed silica-PNIPAM core-shell particles and a PNIPAM nanogel below the radiation-damage threshold are determined. The XPCS data indicate non-diffusive behaviour, suggesting ballistic and stress-dominated heterogeneous particle motions. These results demonstrate the feasibility of XPCS experiments on radiation-sensitive soft-matter materials at FEL sources and pave the way for future applications at MHz repetition rates as well as ultrafast modes using split-pulse devices.