  Modular and mock modular forms possess many striking $p$-adic properties, as
studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori
developed a geometric theory of harmonic Maass forms arising from the de Rham
cohomology of modular curves. In the setting of over-convergent $p$-adic
modular forms, Candelori and Castella showed this leads to $p$-adic analogs of
harmonic Maass forms.
  In this paper we take an analytic approach to construct $p$-adic analogs of
harmonic Maass forms of weight $0$ %and $1/2$
  with square free level. Although our approaches differ, where the two
theories intersect the forms constructed are the same. However our analytic
construction defines these functions on the full super singular locus as well
as on the ordinary locus.
  As with classical harmonic Maass forms, these $p$-adic analogs are connected
to weight $2$ cusp forms and their modular derivatives are weight $2$ weakly
holomorphic modular forms. Traces of their CM values also interpolate the
coefficients of half integer weight modular and mock modular forms. We
demonstrate this through the construction of $p$-adic analogs of two families
of theta lifts for these forms.
