It might just be the announcer's dialect. There are things I routinely "nativize" when I speak English, and I feel little compunction to fret over how a different speech community says them. Unstressed "ay" isn't common in American English and usually defaults to "ee" (so the last syllable of "Sunday" rhymes with "coffee" for a lot of folk, or has a primary and secondary stress-- 'cof-fee but 'Sun,day).
It might be that the guy consulted a different source:
http://dictionary.reference.com/browse/dalai%20lama nicely advises that while the site itself suggests "dah-lahy lahmuh", presumably with good low central "a" in the first three syllables (with that vowel + "y" not being a standard American English diphthong), they also list the American Heritage dictionary with "dä'lī lä'mə". If that doesn't come through on your browser, the first vowel, ä, is the 'a' in 'at' or 'cat', and in many English dialects has routinely raised to an 'eh' or 'ey' sound in some positions. In any event, "dä'lī" is absurdly easy to confuse with "deli" when you're not expecting it. I've only ever heard the not very natural sounding "'dah-,lahy 'lah-muh", where ' = primary stress and , = secondary stress. English doesn't have an unstressed "ahy" diphthong, at least not any American English accent I've run across.
Wikipedia says "dalai lama" is currently pronounced "ta:l: lama" in Tibetan, which seems strange to me: the sequence l: l is presumably not something people would routinely pronounce, so I have to wonder if it's accurate (Ed Garrett, you there?). "Dalai" is apparently an older transliteration of a thoroughly archaic orthography. So we can annoy our friends and enemies by insisting on "Taal lama", and annoy language typologists by treble articulation of the "l": "taalllama" (that's one triple-length "l", not a Babe-like "la-la-la(-la)").