  For quantum symmetric pairs $(\mathbf{U}, \textbf{U}^\imath)$ of Kac-Moody
type, we construct $\imath$canonical bases for the highest weight integrable
$\mathbf{U}$-modules and their tensor products regarded as
$\mathbf{U}^\imath$-modules, as well as an $\imath$canonical basis for the
modified form of the $\imath$quantum group $\mathbf{U}^\imath$. A key new
ingredient is a family of explicit elements called $\imath$divided powers,
which are shown to generate the integral form of $\dot{\bf{U}}^\imath$. 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 $\imath$canonical bases, by avoiding a
case-by-case rank one analysis and removing the strong constraints on the
parameters in a previous work.
