We introduce a new p-adic Maass-Shimura operator acting on a space of "generalized p-adic modular forms" (extending Katz's notion of p-adic modular forms) defined on the p-adic (preperfectoid) universal cover of Shimura curves. Using this operator, we construct new p-adic L-functions in the style of Katz, Bertolini-Darmon-Prasanna and Liu-Zhang-Zhang for Rankin-Selberg families over imaginary quadratic fields K in the case where p is inert or ramified in K. We also establish new p-adic Waldspurger formulas, relating p-adic logarithms of elliptic units and Heegner points to special values of these p-adic L-functions.