Hi Gregory,
thanks a lot. Indeed, now it works:
mr = Import["~/Desktop/cowfix4.stl", "BoundaryMeshRegion"];
uif = NDSolveValue[{Inactive[Laplacian][
u[x, y, z], {x, y, z}] == -0.01,
DirichletCondition[u[x, y, z] == 1, z <= 0.08]},
u, {x, y, z} \[Element] mr];
Needs["NDSolve`FEM`"]
ElementMeshSurfacePlot3D[uif, Boxed -> False, ViewPoint -> {0, -4, 2}]

I noticed that you had to trim the cow's tail.
Thanks,
Marco