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