  Cosmological scenarios built upon the generalized non-local String Field
Theory and $p$-adic tachyons are examined. A general kinetic operator involving
an infinite number of derivatives is studied as well as arbitrary parameter
$p$. The late time dynamics of just the tachyon around the non-perturbative
vacuum is shown to leave the cosmology trivial. A late time behavior of the
tachyon and the scale factor of the FRW metric in the presence of the
cosmological constant or a perfect fluid with $w>-1$ is constructed explicitly
and a possibility of non-vanishing oscillations of the total effective state
parameter around the phantom divide is proven.
