  The metric Markov cotype of barycentric metric spaces is computed, yielding
the first class of metric spaces that are not Banach spaces for which this
bi-Lipschitz invariant is understood. It is shown that this leads to new
nonlinear spectral calculus inequalities, as well as a unified framework for
Lipschitz extension, including new Lipschitz extension results for CAT(0)
targets. An example that elucidates the relation between metric Markov cotype
and Rademacher cotype is analyzed, showing that a classical Lipschitz extension
theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.
