  We prove a general formula for the $p$-adic heights of Heegner points on
modular abelian varieties with potentially ordinary (good or semistable)
reduction at the primes above $p$. The formula is in terms of the cyclotomic
derivative of a Rankin-Selberg $p$-adic $L$-function, which we construct. It
generalises previous work of Perrin-Riou, Howard, and the author, to the
context of the work of Yuan-Zhang-Zhang on the archimedean Gross-Zagier formula
and of Waldspurger on toric periods. We further construct analytic functions
interpolating Heegner points in the anticyclotomic variables, and obtain a
version of our formula for them. It is complemented, when the relevant root
number is $+1$ rather than $-1$, by an anticyclotomic version of the
Waldspurger formula. When combined with work of Fouquet, the anticyclotomic
Gross-Zagier formula implies one divisibility in a $p$-adic Birch and
Swinnerton-Dyer conjecture in anticyclotomic families. Other applications
described in the text will appear separately.
