One can show the short exact sequence by proving that a field is a semisimple ring—alhough one could argue that somewhere hidden in that proof a basis is considered...

I cannot imagine a negative answer to your question in any other form than a model of ZF without the axiom of choice, so that there are vector spaces without bases, and in which maybe short exact sequences of vector spaces which do not split.

One can show the short exact sequence by proving that a field is a semisimple ring—alhough one could argue that somewhere hidden in that proof a basis is considered...

I cannot imagine a negative answer to your question in any other form than a model of ZF without the axiom of choice, so that there are vector spaces without bases, and in which maybe there are short exact sequences of vector spaces which do not split.