We have used the Ultra-High-Resolution Facility (UHRF) at the AAT, operating at a resolution of 0.35 km s(-1) (FWHM), to observe K I and C(2) absorption lines arising in the circumstellar environment of the post-AGB star HD 56126. We find three narrow circumstellar absorption components in K I, two of which are also present in C(2). We attribute this velocity structure to discrete shells resulting from multiple mass-loss events from the star. The very high spectral resolution has enabled us to resolve the intrinsic linewidths of these narrow lines for the first time, and we obtain velocity dispersions (b-values) of 0.2-0.3 km s(-1) for the K I components, and 0.54 +/- 0.03 km s(-1) for the strongest (and best defined) C(2) component. These correspond to rigorous kinetic temperature upper limits of 211 K for K I and 420 K for C(2), although the b-value ratio implies that these two species do not co-exist spatially. The observed degree of rotational excitation of C(2) implies low kinetic temperatures (T(k) approximate to 10 K) and high densities (n approximate to 10(6) to 10(7) cm(-3)) within the shell responsible for the main C(2) component. Given this low temperature, the line profiles then imply either mildly supersonic turbulence or an unresolved velocity gradient through the shell.