Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$
See two papers by Kubota: Ein arithmetischer Satz über eine Matrizengruppe (1966, MR0188194) and Topological Covering of SL(2) Over a Local Field (1967, MR0204422). Maybe these are the first references, although in isome sense it goes back (at least) to Weil's famous Acta paper Sur certains groupes d'opérateurs unitaires (1964, MR0165033).