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In the present paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in L 2 ( Q p n ) is constructed. These bases cannot be constructed in the framework of any of known theories. We use the wavelet bases in the following applications: in the theory of p-adic pseudo-differential operators and equations. The connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators is studied. We derive a criterion for a multidimensional p-adic wavelet function to be an eigenfunction for a pseudo-differential operator and prove that our wavelets are eigenfunctions of the fractional operator. p-Adic wavelets are used to construct solutions of linear (the first and second order in t) and semi-linear evolutionary pseudo-differential equations. Since many p-adic models use pseudo-differential operators (fractional operator), our results can be intensively used in these models.