  We discuss canonical local heights on abelian varieties over non-archimedean
fields from the point of view of Berkovich analytic spaces. Our main result is
a refinement of N\'eron's classical result relating canonical local heights
with intersection multiplicities on the N\'eron model. We also revisit Tate's
explicit formulas for N\'eron's canonical local heights on elliptic curves. Our
results can be viewed as extensions of Tate's formulas to higher dimensions.
