Does age play a role in language acquisition?

Yes. But contrary to popular belief, the older you are, the faster you acquire a language.

In short, this is because the brains of babies and children are still in development. Plus adults know many of the concepts (both grammatical but also just what words point to in general - things, phenomenon etc.).

Furthermore, adults can compliment language acquisition methods with language learning methods. Young children aren’t yet able to do it.

Studies

Age, Rate and Eventual Attainment in Second Language Acquisition on JSTOR

Stephen D. Krashen, Michael A. Long, Robin C. Scarcella, Age, Rate and Eventual Attainment in Second Language Acquisition, TESOL Quarterly, Vol. 13, No. 4 (Dec., 1979), pp. 573-582