Katz type $p$-adic $L$-functions for primes $p$ non-split in the $\mathrm{CM}$ field

For every triple F , K , p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p is an odd prime integer which is not split in K , we attach p -adic L -function which interpolates the algebraic parts of the special values of the complex L -functions of F twisted by algebraic Hecke characters of K such that the p -part of their conductor is p^{n} , with n large enough (for p\geq 5 it suffices n\ge 2 ). This construction extends a classical construction of N. Katz for F an Eisenstein series, and of Bertolini–Darmon–Prasanna for F a cuspform when p is split in K . Moreover, we prove a Kronecker limit formula, respectively, p -adic Gross–Zagier formulae, for our newly defined p -adic L -functions.