We investigate the properties of the fine structure of a sunspot penumbra based on spectroscopic measurements with high spectral (λ/δλ = 250 000) and high spatial (≈0.5 arcsec) resolution. The magnetically insensitive Fe I 557.6 nm line is used to probe the penumbral atmosphere. The data was taken at the German Vacuum Tower Telescope with the 2D-spectrometer TESOS, taking advantage of the recently installed Kiepenheuer Adaptive Optics System (KAOS). The field of view covers a sunspot located at 23 ◦ off the disk center and its immediate surroundings. The penumbral structure is studied by means of maps computed for the line-of-sight velocity, the line width, the equivalent width and the line depression. Line-of-sight velocities are derived from the Doppler shifts at different bisector levels. From these maps we infer the flow field geometry and study the azimuthal and radial dependences of the line parameters. Our findings can be summarized as follows: (a) the flow pattern has a conspicuous filamentary structure in the deep photospheric layers and is rather diffuse in the high layers. (b) The flow field slightly spreads and fans out with height. (c) The flow geometry confirms the presence of an upflow component in the inner penumbra and a downflow component in the middle and outer penumbra. (d) We find an enhanced brightness of the mid-penumbra (bright ring) in the line wings, but not in the continuum or line core. (e) The azimuthal average of the equivalent width, the line width and the absolute flow velocity increase with radial distance within the penumbra. (f) Small- scale variations of the equivalent width and the line width on the center-side penumbra are co-spatial and correlated with (blue-shifted) fluctuations in the line-of-sight velocity. (g) Inner limb-side penumbral grains are associated with blue-shifts of v ≤− 400 m s −1 , indicating upflows. (h) One umbral dot in our sample is associated with a blue-shift of v = −200 m s −1 .