Let an Infinity Axiom be any first-order sentence $\varphi$ such that for any model $M$, if $M \models \varphi$, then the universe of $M$ has to be infinite.
An example of such a sentence is a $\forall\exists$-formula over the signature consisting of one unary function symbol, say $f$, asserting that $f$ is a non-surjective injection.
It is well-known that in a functional language consisting of one unary function symbol, say $S$, and one constant (i.e. $0$-ary function symbol), say $z$, there is a $\forall$-formula (i.e., a purely universal one) that is an Infinity Axiom, according to the definition above.
Question: if we restrict ourselves to a signature that consists only of one (unary, or binary, whatever), function symbol, does there exist a purely universal formula that is an Infinity Axiom? If not, how can one prove it?