Hi Boris, and welcome to the forum!
All the dictionaries I have show smile as a one-syllable word. But I see your quandary. There are many words in English that are "borderline" in that respect. Smile is one of many. In some regional dialects, it probably does get pronounced as a two-syllable word. But you'll never see it hyphenated as "smi-le" unless you force that line-break.
Here is a pronunciation dictionary version of smile.
It is officially one syllable, but I pronounce it, along with mile, file, pile, etc., as a rhyme to trial, which officially has two syllables. I'm sure there are many dialects that pronounce each of these and trial as one syllable.
It is not uncommon for the official syllabification and the actual pronunciation to be significantly different: Wednesday; interesting; athlete; laboratory.
I'm rereading Ron Luciano's The Umpire Strikes Back. Luciano was a pro football player who after his third injury discovered umpiring baseball. He describes Emmett Ashford as "major league baseball's first black umpire….a wonderful showman and the only umpire I've ever known capable of turning 'ball' into a six-syllable word."
I add "comfortable" and "vegetable" to Glenn's list. In the US (though not in the UK), "vegetable" is pronounced with only three syllables, "VEHJ-teh-bl". We do "comfortable" in three, too: "CUMF-tr-bl". Note the movement of the 'r' in the latter. Oh, and "probably" is usually "PRAHB-lee", though not always.
Anyway, I'm with the rest of you on "smile", and the same with "child", "lyre", "rowl" and "flour". Do you see the common factor? In each case, an 'r' or 'l' follows a dipthong ending in 'ee' or 'oo'. Apparently our physiology is such that we almost have to pronounce those as separate syllables.
(A diphthong is a sliding combinations of two vowel sounds, mostly "eye" and "ow" but there are many others. In the word "sigh", the vowel sound is actually two vowels, an 'ah' sliding into 'ee'. In "cow", the 'ah' sound slides in the opposite direction, into 'oo'. The most common dipthongs slide toward 'ee' or 'oo'. Physiologically "are" and "ill" are diphthongs, too, where the starting sound slides into the vowels 'r' and 'l', but they're not traditionally classed that way.)
"Buyer" and "byre" are pronounced exactly the same, but one is classified as two syllables and the other as one. Same with "flower" and "flour". At the other end, "dial" is two syllables but "bile" is one; likewise "towel" and "jowl". The dictionaries may say those are one-syllable words, but in my opinion the only way to pronounce them so is to short the 'ee' or 'oo'; if you manage "fire" as a one-syllable word, you're probably from Georgia.
(I meant that as a joke, but for non-Yankees I should explain that in the American southeast (not just Georgia) the regional accent does indeed involve cutting diphthongs short; "fire" sounds a bit like "far", "oil" like "ole" and so forth.)
Glenn, your posts today and back in July interestingly go together to describe me and many other Texans. I say "smile" exactly as you describe with one syllable and even though this is not exactly what you meant when you said "smile" could rhyme with "trial", that is how I say it, both with one syllable. The only exception is if I'm trying to think of what to say. Then either one could have 2, 3, maybe 4 syllables.