  Westerlund 1 (Wd 1) is the most massive stellar cluster in the Galaxy and
associated with an extended region of TeV emission. Here we report the results
of a search for GeV gamma-ray emission in this region. The analysis is based on
~4.5 years of Fermi-LAT data and reveals significantly extended emission which
we model as a Gaussian, resulting in a best-fit sigma of sigma_S = (0.475 +/-
0.05) deg and an offset from Wd 1 of ~1 deg. A partial overlap of the GeV
emission with the TeV signal as reported by H.E.S.S. is found. We investigate
the spectral and morphological characteristics of the gamma-ray emission and
discuss its origin in the context of two distinct scenarios. Acceleration of
electrons in a Pulsar Wind Nebula provides a reasonably natural interpretation
of the GeV emission, but leaves the TeV emission unexplained. A scenario in
which protons are accelerated in or near Wd 1 in supernova explosion(s) and are
diffusing away and interacting with molecular material, seems consistent with
the observed GeV and TeV emission, but requires a very high energy input in
protons, ~10^51 erg, and rather slow diffusion. Observations of Wd 1 with a
future gamma-ray detector such as CTA provide a very promising route to fully
resolve the origin of the TeV and GeV emission in Wd 1 and provide a deeper
understanding of the high-energy (HE) astrophysics of massive stellar clusters.
