Actually, according to the article on the Dalai Lama:
The name "Dalai Lama" is a combination of the Mongolic word dalai meaning "ocean" or "great" (coming from Mongolian title Dalaiyin qan or Dalaiin khan, translated as Gyatso or rgya-mtsho in Tibetan) and the Tibetan word བླ་མ་ (bla-ma) meaning "master, guru".
The Buddhist Lama and the Assyrian Lama just happen to be two words from different languages that happen to sound the same, no more related to each other than either is to the animal classification of llama.
Actually, according to the article on the Dalai Lama:
The Buddhist Lama and the Assyrian Lama just happen to be two words from different languages that happen to sound the same, no more related to each other than either is to the animal classification of llama.