  We show that every geodesic metric space admitting an injective continuous
map into the plane as well as every planar graph has Nagata dimension at most
two, hence asymptotic dimension at most two. This relies on and answers a
question in a very recent work by Fujiwara and Papasoglu. We conclude that all
three-dimensional Hadamard manifolds have Nagata dimension three. As a
consequence, all such manifolds are absolute Lipschitz retracts.
