  A variety of different orthogonal wavelet bases has been found in L_2(R) for
the last three decades. It appeared that similar constructions also exist for
functions defined on some other algebraic structures, such as the Cantor and
Vilenkin groups and local fields of positive characteristic. In the present
paper we show that the situation is quite different for the field of $p$-adic
numbers. Namely, it is proved that any orthogonal wavelet basis consisting of
band-limited (periodic) functions is a modification of Haar basis. This is a
little bit unexpected because from the wavelet theory point of view, the
additive group of $p$-adic numbers looks very similar to the Vilenkin group
where analogs of the Daubechies wavelets (and even band-limited ones) do exist.
We note that all $p$-adic wavelet bases and frames appeared in the literature
consist of Schwartz-Bruhat functions (i.e., band-limited and compactly
supported ones).
