  In this paper a countable family of new compactly supported {\em non-Haar}
$p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ is constructed. We use the wavelet
bases in the following applications: in the theory of $p$-adic
pseudo-differential operators and equations. Namely, we study the connections
between wavelet analysis and spectral analysis of $p$-adic pseudo-differential
operators. A criterion for a multidimensional $p$-adic wavelet to be an
eigenfunction for a pseudo-differential operator is derived. We prove that
these wavelets are eigenfunctions of the fractional operator. In addition,
$p$-adic wavelets are used to construct solutions of linear and semi-linear
pseudo-differential equations. Since many $p$-adic models use
pseudo-differential operators (fractional operator), these results can be
intensively used in these models.
