The answer is "almost no" - the wavelength of the photon is virtually unchanged (in the initial rest frame of the mirror). Because the mirror is much more "massive" than the photon, it serves as a "momentum sink" and picks up almost no energy.
The best way to develop an intuition for this is to consider a collision between two balls: one lighter (with mass $m$) and initially moving (at velocity $v_1$) and one more massive (with mass $M$) and initially at rest. After scattering, the lighter ball leaves the scene at velocity $v_3$ and the more massive ball leaves the scene at velocity $v_4$.
Set $v_2 = 0$ in the following worked-out example (see page 3): http://www.its.caltech.edu/~teinav/Lectures/Ph%201a/Lecture%207%20-%202017-10-19.pdf
In the limit that $M >> m$ the amount of kinetic energy picked up by the massive object goes to zero, but it acquires twice as much momentum (and in the opposite direction) as the lighter object initially had. Thusly can momentum be transferred but (almost) no energy.