  We present the first temperature surface map of EK Dra from
very-high-resolution spectra obtained with the Potsdam Echelle Polarimetric and
Spectroscopic Instrument (PEPSI) at the Large Binocular Telescope. Changes in
spectral line profiles are inverted to a stellar surface temperature map using
our $i$Map code. The long-term photometric record is employed to compare our
map with previously published maps. Four cool spots were reconstructed, but no
polar spot was seen. The temperature difference to the photosphere of the spots
is between 990 and 280K. Two spots are reconstructed with a typical solar
morphology with an umbra and a penumbra. For the one isolated and relatively
round spot (A), we determine an umbral temperature of 990K and a penumbral
temperature of 180K below photospheric temperature. The umbra to photosphere
intensity ratio of EK Dra is approximately only half of that of a comparison
sunspot. A test inversion from degraded line profiles showed that the higher
spectral resolution of PEPSI reconstructs the surface with a temperature
difference that is on average 10% higher than before and with smaller surface
areas by 10-20%. PEPSI is therefore better suited to detecting and
characterising temperature inhomogeneities. With ten more years of photometry,
we also refine the spot cycle period of EK Dra to 8.9$\pm$0.2 years with a
continuing long-term fading trend. The temperature morphology of spot A so far
appears to show the best evidence for the existence of a solar-like penumbra
for a starspot. We emphasise that it is more the non-capture of the true umbral
contrast rather than the detection of the weak penumbra that is the limiting
factor. The relatively small line broadening of EK Dra, together with the only
moderately high spectral resolutions previously available, appear to be the
main contributors to the lower-than-expected spot contrasts when comparing to
the Sun.
