We present integral field spectroscopic observations with the Potsdam Multi-Aperture Spectrophotometer of all 14 galaxies in the z ~ 0.1 Lyman Alpha Reference Sample (LARS). We produce 2D line-of-sight velocity maps and velocity dispersion maps from the Balmer α (Hα) emission in our data cubes. These maps trace the spectral and spatial properties of the LARS galaxies’ intrinsic Lyα radiation field. We show our kinematic maps that are spatially registered onto the Hubble Space Telescope Hα and Lyman α (Lyα) images. We can conjecture a causal connection between spatially resolved Hα kinematics and Lyα photometry for individual galaxies, however, no general trend can be established for the whole sample. Furthermore, we compute the intrinsic velocity dispersion σ0, the shearing velocity vshear, and the vshear/σ0 ratio from our kinematic maps. In general LARS galaxies are characterised by high intrinsic velocity dispersions (54 km s-1 median) and low shearing velocities (65 km s-1 median). The vshear/σ0 values range from 0.5 to 3.2 with an average of 1.5. It is noteworthy that five galaxies of the sample are dispersion-dominated systems with vshear/σ0< 1, and are thus kinematically similar to turbulent star-forming galaxies seen at high redshift. When linking our kinematical statistics to the global LARS Lyα properties, we find that dispersion-dominated systems show higher Lyα equivalent widths and higher Lyα escape fractions than systems with vshear/σ0> 1. Our result indicates that turbulence in actively star-forming systems is causally connected to interstellar medium conditions that favour an escape of Lyα radiation.