No. In fact the opposite is the case.
It is a common misbelief that galaxies receding faster than the speed of light are not visible to us. This is not the case; we easily see galaxies moving at superluminal velocities. This does not — as I think most people would think — contradict the theory relativity, since nothing travel through space faster than $c$.
We see "super-luminal" galaxies
The recession velocity $v_mathrm{rec}$ of a galaxy is given by Hubble's Law:
$$
v_mathrm{rec} = H_0 d,
$$
where $H_0 simeq 67.8,mathrm{km},mathrm{s}^{-1},mathrm{Mpc}^{-1}$ is the Hubble constant. This law implies that galaxies farther away than
$$
d_{v>c} = c/H_0 = 4400,mathrm{Mpc} = 14.4 , mathrm{Gly}
$$
recede faster than $c$ ("Gly" means giga-lightyears). Objects at this distance have a redshift of $zsimeq1.5$.
Consider a photon emitted from a distant galaxy (say, GN-z11 at redshift $z=11.1$) in the past, in the direction of the Milky Way. What special relativity tells us is that locally, the photon always travels through space at $v=c$. Initially, the photon thus increases it distance from GN-z11 at velocity $c$. However, even though the photon travels toward us, its distance to MW increases, due to the expansion of the Universe. As the photon increases its distance to GN-z11, the same expansion causes it to recede from GN-z11 at an ever-increasing velocity. Moreover, as it travels toward MW, it will slowly "overcome" the expansion until it reaches the point where $v_mathrm{rec} = c$. For an infinitesimally small period, it will stand will wrt. MW, after which it will begin to travel faster and faster as measured from MW. Eventually, its velocity — still in MW's reference frame — will reach $c$, at which point it will have reached MW.
Thus, even though GN-z11 and MW recede from each other at $v_mathrm{rec} = 2.2c$, we are still able to see it.
We see more and more distant galaxies
There is, however, a limit to how fast a galaxy visible to us can recede, given by the distance $d_mathrm{PH}$ that light has had the time to travel since the Universe was created. Light comes to us from all directions, so we're situated in the center of a sphere of radius $d_mathrm{PH}$. This sphere is called "the observable Universe", and its surface (which is not a physical thing) is called the particle horizon (hence the subscript "PH"). Galaxies at the particle horizon are receding at $v_mathrm{rec}simeq3.3c$.
As time goes by, light from ever-more-distant galaxies$^dagger$ will reach us; that is $d_mathrm{PH}$ increases. In other words, the observable Universe always increases in size, and no galaxy visible today will ever leave the observable Universe, no matter its speed.
However, since future observable galaxies will be more and more redshifted, their light will eventually shift out of the visible range and into longer and longer radiowaves. Furthermore, the time between each detected photon will increase, so they will be dimmer and dimmer, and thus in practice, they will disappear.
$^dagger$Note that since large distances also means looking back in time (since the light has spent a long time traveling), we actually don't see galaxies this far away, as they hadn't formed this early in history. We do however see the gas from which the galaxies were born, as far back as 380,000 years after Big Bang.
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