This is from Somerset Maugham’s “The escape”. "She (Mrs. Barlow) was apparently one of those unfortunate people with whom nothing by any chance goes right. If she married a husband he beat her; if she employed a broker he cheated her; if she engaged a cook she drank. She never had a little lamb but it was sure to die.” This means to say that she never had a lamb but if she had a lamb, it was sure to die. i.e. nothing went right with her. the reason why the author has chosen "Little lamb" is because it is part of a very famous nursery rhyme "Mary had a little lamb". Another example : he never sat in an exam but he was sure to fail. i.e. he was so bad at studies that he would have failed if he had ever sat in any exam.

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It's not an idiom. It is taken from a well known nursery rhyme: Mary had a little lamb...... This is Somerset Maugham, right? Everything dear to her eventually died. In fact, there was nothing dear to her that did not die. That is the meaning of the phrase.