This article is the first in a series devoted to studying generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and the Euler system properties of the associated Selmer es, with special emphasis on their application to the Birch-Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. The basis for the entire study is a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of certain $p$-adic $L$-functions attached to the Garrett-Rankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula.