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For a prime number $p$, we study the asymptotic distribution of CM points onthe moduli space of elliptic curves over $\mathbb{C}_p$. In stark contrast tothe complex case, in the $p$-adic setting there are infinitely many differentmeasures describing the asymptotic distribution of CM points. In this paper weidentify all of these measures. A key insight is to translate this problem intoa $p$-adic version of Linnik's classical problem on the asymptotic distributionof integer points on spheres. To do this translation, we use the closerelationship between the deformation theories of elliptic curves and formalmodules and then apply results of Gross and Hopkins. We solve this $p$-adicLinnik problem using a deviation estimate extracted from the bounds for theFourier coefficients of cuspidal modular forms of Deligne, Iwaniec and Duke. Wealso identify all accumulation measures of an arbitrary Hecke orbit.

p-Adic distribution of CM points and Hecke orbits. II: Linnik equidistribution on the supersingular locus