This question already has an answer here:
Breaking of a local symmetry is impossible. It is often said that therefore the role of the Higgs mechanism in the standard model is a different one.
Once a gauge is fixed, however, to remove the redundant degrees of freedom, the remaining (discrete!) global symmetry may undergo spontaneous symmetry-breaking exactly along the lines discussed in the previous chapter. The phrase "spontaneous breaking of local gauge symmetry" is therefore in some sense a misnomer, but a convenient one, if we think of it as a short circumlocution for "spontaneous breaking of remnant global symmetry after removal of redundant gauge degrees of freedom by appropriate gauge fixing".
Or here's a similar statement from a different source:
But is the gauge symmetry actually broken spontaneously? In the above exposition of the Higgs mechanism, there were two instances when a symmetry was broken. First, when we selected one minimum out of infinite amount of equivalent minima, a spontaneous breaking indeed took place, but only of a global symmetry. This minimum represents a vacuum, and in order to perturbatively describe the quantum field theory, we need to quantize the fields. Quantization of gauge field theories requires introduction of a gauge-fixing procedure, and during this procedure we break the gauge symmetry by hand, explicitly, not spontaneously. Thus, the two notions, EWSB and SSB, are in certain sense correct, but they do not refer to the same symmetry. [...] As Englert says in his Nobel lecture : “… The vacuum is no more degenerate and strictly speaking there is no spontaneous symmetry breaking of a local symmetry. The reason why the phase with nonvanishing scalar expectation value is often labeled SSB is that one uses perturbation theory to select at zero coupling with the gauge fields a scalar field configuration from global SSB; but this preferred choice is only a convenient one.
What global symmetry are they referring to? (I find it extremely strange that they don't specify the allegedly broken global symmetry.)