Canonical bases arising from quantum symmetric pairs of Kac-Moody type

For quantum symmetric pairs $(\mathbf{U},{\mathbf{\text{U}}}^{\mathrm{ı}})$ of Kac-Moody type, we construct $\mathrm{ı}$canonical bases for the highest weight integrable $\mathbf{U}$-modules and their tensor products regarded as ${\mathbf{U}}^{\mathrm{ı}}$-modules, as well as an $\mathrm{ı}$canonical basis for the modified form of the $\mathrm{ı}$quantum group ${\mathbf{U}}^{\mathrm{ı}}$. A key new ingredient is a family of explicit elements called $\mathrm{ı}$divided powers, which are shown to generate the integral form of ${\dot{\mathbf{U}}}^{\mathrm{ı}}$. We prove a conjecture of Balagovic-Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-K-matrix and the constructions of $\mathrm{ı}$canonical bases, by avoiding a case-by-case rank one analysis and removing the strong constraints on the parameters in a previous work.