First-order mutual-recursive functions Turing-complete or incomplete?

Suppose we have an ML-like programming language with only first-order terms (i.e. no higher-order functions/lambdas; variables cannot be functions). However, the language allows recursion in all forms.

Is it true that this language is Turing-incomplete, but becomes complete if we add basic heap semantics (i.e. pointers and manipulation of RAM-like memory)?