TDD approach issue with test

I'm new to TDD practice and i was wondering if my approach is good from a special case.

I want to program a little software that takes a group of array strings in parameter and split all string in that group.

Example :

{"C - 3 - 3", "M - 5 - 5"} should return { {"C", "3", "3"}, {"M", "5", "5"} }

By cutting out the problem, i started with a StringSplitter and StringSplitterTest with TDD in order to split just one string to an array of string.

After, i programmed a StringGroupSplitter and StringGroupSplitterTest (always with the TDD approach) doing the same thing but with a array of strings (knowing that StringGroupSplitter has a StringSplitter dependency).

So, i remembered the "FIRST" principles of unit tests and in particular the principle of independence saying that a test should not depend on the result of another test.

In my case, the problem is that due to the dependency of StringGroupSplitter and StringSplitter, if only ont test fails in StringSplitterTest, tests in StringGroupSplitterTest will also fail.

So my question is the following : Is my approach to TDD the right one ?

Thanks in advance.