In the theory of $p$-adic $L$-functions a $p$-adic Gross-Zagier formula gives interpretation to special values of $p$-adic $L$-functions outside the region of interpolation using $p$-adic integration. Seen as a first step towards “explicit reciprocity laws” they have important applications towards proving various instances of the Bloch-Kato conjecture, as in the work of Darmon–Rotger. We construct a $p$-adic twisted triple product $L$-function associated to finite slope families of Hilbert modular forms, assuming $p$ unramified in the totally real fields. We use techniques of $p$-adic iteration of the Gauss–Manin connection on sheaves of nearly overconvergent modular forms, as developed by Andreatta–Iovita. In joint work with Ting-Han Huang, we prove a $p$-adic Gross-Zagier formula for this $L$-function for a pair of an elliptic modular form and a quadratic Hilbert modular form. This generalises work of Blanco-Chacon and Fornea for the case of Hida families, and we overcome a technical assumption in their work of $p$ being split in the quadratic field.