Can a photon have a wavelength less than the planck length?

If there was a minimum wavelength, you just could increase your own velocity in direction of the photon to still make it smaller.

Since the relative velocity to the photon must always be c the only thing that can increase then is the frequency, so you always can get a still smaller wavelength just by increasing the doppler effect.

Because of that there is no minimum wavelength. If you could reach c (which you can't) the wavelength would be zero, but since you can get arbitrarily close to c the wavelength can also get arbitrarily close to zero.

I am going to weigh in on this. In one sense we can say that we really do not know. Since you mentioned experimental evidence, we have no experimental evidence of anything at the Planck energy $E~=~\sqrt{c^5/G\hbar}$. The wave length of a particle at this energy would be equal to the Planck length. The Planck length is computed by equating the wavelength of a particle at rest with the circumference of the event horizon. The $4$-momentum $P^\mu~=~(mc,~0,~0,~0)$ with the deBroglie type equation $p\lambda~=~h$, we equation $\lambda$ equated to $2\pi$ times the Schwarzschild radius $r~=~2GM/c^2$ the rest is algebra and you get $\ell_p~=~\sqrt{G\hbar/c^3}$

So what does it mean for a particle to have a wavelength shorter than $\ell_p$? It means the particle is in a region smaller than the even horizon of the smallest quantum unit of black hole. There is nothing immediate that says this can't happen. What we do hypothesize is that this unit of black hole represents the smallest region one can locate a qubit. 't Hooft and Susskind formulated holography initially by looking at event horizons according to units of Planck areas that can hold a qubit of information.

So suppose you have photons or any quantum field with an arbitrary spectrum of energy or frequencies. The energy in a Fourier sum that exceeds the Planck energy is not able to hold a qubit of information. In other words, these probably do not play any physically meaningful role and can be removed. This follows in some sense with the idea that the Planck scale is sort of the ultimate renormalization cut-off in QFT or quantum gravity. Of course as yet the details of this are not yet certain.

The short answer is we simply do not know: this hypothesis is utterly beyond anything that we can either test experimentally or reason about with a widely accepted theory.

Симон Тыран's Answer is a good, concise exposition showing what classical, relativistic reasoning has to say about this. But I don't believe this answers the question because you have to make the assumption that classical relativity works down to an arbitrarily small scale and that's something we neither know nor (I get the impression as a lay reader) even believe.

Moreover, even without full quantum gravity, one can formulate classical theories with both an invariant velocity (as in the $c$ of special relativity) and and invariant length scale. Examples of such theories are "Doubly Special Relativity" and also de Sitter Invariant Special Relativity wherein the symmetry group $SO(1,\,4)$ is a supergroup of the Lorentz group and is the same as the symmetry group of de Sitter space, a highly symmetric vacuum solution of the Einstein field equations. In such a universe one would have a natural, invariant length scale that can be used to invalidate the classical special relativistic reasoning that there always exists an inertial observer whom a wavelength is arbitrarily small for.