Can you get drunk on ginger beer?

Yes.

Ginger Beer is made from ginger, water, sugar, lemon juice and a rather special bacterial-fungal symbiote known, unsurprisingly, as the Ginger Beer Plant. It ferments up to 11% alcohol.

However, most things called ginger beer today are soft drinks, and i don't think there's a single alcoholic Ginger Ale on the market. The difference between ginger ale and ginger beer is huge.