first order logic – Downward Lowenheim Skolem over an uncountable vocabulary

Downward Lowenheim Skolem says that there exists even a countable model for a set F of first order logic formulas, if

  • the vocabulary and variables are countable
  • F is satisfiable

Can one show a counterexample for the case where the assumption countable vocabulary is not fulfilled,
i.e. give an uncountable vocabulary together with a formula set F over this vocabulary, such that F is satisfiable but has no countable model?