Let Ω be a bounded pseudoconvex domain in ${ℂ}^n$ with $C^1$ boundary and let X be a complete intersection submanifold of Ω, defined by holomorphic functions $v_1,...,v_p$ (1 ≤ p ≤ n-1) smooth up to ∂Ω. We give sufficient conditions ensuring that a function f holomorphic in X (resp. in Ω, vanishing on X), and smooth up to the boundary, extends to a function g holomorphic in Ω and belonging to a given strongly non-quasianalytic Carleman class $l!M_l$ in $\overline{\Omega }$ (resp. satisfies $f=v_1{f}_1+...+v_p{f}_p$ with $f_1,...,f_p$ holomorphic in Ω and $l!M_l$-regular in $\overline{\Omega }$). The essential...