Let E be a (semistable) rational elliptic curve of conductor N, let K be an imaginary quadratic field satisfying a “Heegner hypothesis” relative to N and let p be a prime of split multiplicative reduction for E that splits in K. Following a recipe proposed by Bertolini and Darmon, I will define a p-adic L-function L_p(E/K) in terms of distributions of Heegner points on Shimura curves that are rational over the anticyclotomic Z_p-extension of K. The “special value” of L_p(E/K) is 0, and I will sketch a proof of a Gross-Zagier formula for the first derivative of L_p(E/K) involving a Heegner point over K and a p-adic L-invariant of E à la Mazur-Tate-Teitelbaum. The strategy is based on level raising arguments and Jochnowitz-type congruences. This is joint work (in progress) with Rodolfo Venerucci.