The Mermin-Wagner theorem states that continuous global symmetries cannot be broken in two or fewer spacetime dimensions; however, I have not seen this statement applied to gauge theories. Does it apply; ie, is there a Higgs mechanism for 1+1 QFTs?
Gauge symmetry is actually not spontaneously broken in the Higgs mechanism; this is a common misconception. See What role does "spontaneously symmetry breaking" played in the "Higgs Mechanism"?. Therefore the Mermin-Wagner theorem does not apply to the Higgs mechanism, and the Higgs mechanism is possible in 1+1D.