I've seen this in a couple of places, and I don't understand where it's coming from.  There are *two* issues with the calculation. The first is that $v=f\lambda$ shouldn't be used *at all*, because that's the dispersion relation for *massless* waves in linear media (like light in vacuum); the relationship between $v$ and $\lambda$ for matter waves is just different (see below).  Furthermore, OP is right that $c$ shouldn't come into it at all, because nothing in a matter waves is actually moving at speed $c$ (whether we're talking about group velocity or phase velocity).  Below the break, some details.

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The relationship $v=f\lambda$ is not the correct expression for matter waves.  To understand things more generally, let's rewrite this as
$$
\omega = v k\,,
$$
where $\omega=2\pi f$ is the angular frequency and $k=2\pi\lambda$ is the wave vector.

The important notion here is known as the *dispersion relation* of the wave. It relates the temporal properties of the wave (i.e., its frequency $\omega$) to the *spatial* properties of the wave (i.e., its wave vector), and this relationship is *different* for different types of wave.  For light in vacuum, the relationship is exactly $\omega=ck$, but for *matter* waves, the relationship is different.

de Broglie's hypothesis relates the energy of a matter wave to its frequency, i.e., $E=hf = \hbar\omega$, and the momentum to its wavelength, i.e., $p = h/\lambda = \hbar k$.

Now, the energy of a free particle of mass $m$ is given by just the kinetic energy, $p^2/2m$.  According to the de Broglie *hypothesis*, the momentum of a particle is given by $p=\hbar k = h/\lambda$, i.e, the momentum of a particle is related to the *spatial* characteristics of the corresponding matter wave. Then, using the kinetic energy of a particle, we can derive the dispersion relation for matter waves as
$$
\hbar\omega = E = \frac{p^2}{2m} = \frac{(\hbar k)^2}{2m}\,,
$$
so that
$$
\omega = \frac{\hbar k^2}{2m}\,.
$$

Now, without getting into the details (you should look these up), the *phase* velocity of a monochromatic wave of angular frequency $\omega$ is given by $v_p=\omega/k$, whereas the *group* velocity of a *wave-packet* centered near $k$ is give by $v_g = d\omega/dk$.  Therefore, for a matter wave,
$$
v_p = \frac{\omega}{k} = \frac{\hbar k^2/2m}{k} = \frac{\hbar k}{2m}
=\frac{h}{2m\lambda}\,,
$$
and
$$
v_g = \frac{d\omega}{dk} = \frac{d}{dk}\frac{\hbar k^2}{2m} = \frac{\hbar k}{m}
=\frac{h}{m\lambda}\,.
$$

Using $v=f\lambda$ is just wrong in this context, and *especially* we shouldn't have a $c$ in there.