A particle does not have a wave function with respect to itself; but for any observer, the uncertainty principle tells us that if a particle could be known to have any exact velocity (and in particular if it is known to be in the observer’s rest frame with a velocity of zero) then its position would be completely unknown – and in the case of zero velocity this would make the wave function constant. (Of course, in practice we never know either the position or the momentum exactly, and this corresponds to the mathematical fact that the constant amplitude wave-function is not normalizable.)
A typical realistic position space wave function is in the form of a wave packet which has an amplitude representing the probability density multiplied by a complex phase factor which oscillates (or more precisely rotates around the unit circle in the complex plane) at a frequency corresponding to the average observed velocity. As the velocity goes to zero, the wavelength of those phase oscillations goes to infinity and the wavefunction just looks like a bump of almost constant phase. But this infinite wavelength does not mean that the wavefunction is constant, and the shape of its amplitude envelope means that its fourier transform includes contributions from frequencies other than that corresponding to the average observed velocity (and so the momentum space wave function is also a bump with width related to that in position space by the uncertainty principle).