  In this paper, we introduce a new type of matter that has origin in $p$-adic
strings, i.e., strings with a $p$-adic worldsheet. We investigate some
properties of this $p$-adic matter, in particular its cosmological aspects. We
start with crossing symmetric scattering amplitudes for $p$-adic open strings
and related effective nonlocal and nonlinear Lagrangian which describes tachyon
dynamics at the tree level. Then, we make a slight modification of this
Lagrangian and obtain a new Lagrangian for non-tachyonic scalar field. {Using
this new Lagrangian in the weak field approximation as a matter in Einstein
gravity with the cosmological constant, one obtains an exponentially expanding
FLRW closed universe.} At the end, we discuss the obtained results, i.e.,
computed mass of the scalar $p$-adic particle, estimated radius of related
closed universe and noted $p$-adic matter as a possible candidate for dark
matter.
