It's possible that the word your mother used was, or was derived from, the word ちんけ (chinke). This is a 形容動詞【けいようどうし】 or na-adjective.
- ちんけな店舗【てんぽ】 (chinke na tenpo, "a tiny shop")
Apparently the "one" on a die was called ち (chi) or ちん (chin), or possibly even ちんけ (chinke), in gambling circles. The adjective ちんけ (chinke) derived from the "one" sense, meaning "tiny, the lowest, the smallest, insignificant". Apparently it also has (had?) a slang meaning of "someone who is blind in one eye".
See more in the Daijisen and Daijirin dictionary entries visible here at the Kotobank aggregator site, or in the Agatsuma Dialect Dictionary and Jargon Dictionary entries visible here at the Weblio aggregator site.
In response to comments added to the question post after I'd written the above:
- She would use it very often to describe anything from small ears to small amount of… あなたの耳はとってもチンキですねかわいいOr to describe something that was smaller than the normal size of the object like miniature eclairs versus regular sized eclairs – Shoga NAGAI
あなたの耳はとってもチンキですねかわいい <-- It doesn't seem to be connected with ちんけ, then... since ちんけ has a negative/derogative connotation.. – Chocolate
As Chocolate notes, the ちんけ term from gambling does have negative overtones, as befitting a description of a low score in a game where high scores are good.
ちんき has several possible kanji spellings. None of them mean "small", which is why I suggested a shift from ちんけ earlier. The poster's ちんき could alternatively be a sense development from 珍奇【ちんき】 ("queer, odd") or 珍稀【ちんき】 ("rare, unusual"), perhaps calling out the exceptionally small size of something. Another (more?) likely possibility is a sense development from 珍貴【ちんき】 ("unusual and precious"). Even in English, "precious" is sometimes used to describe small things.