Let $A$ be an $\mathbf{F}_p$-algebra such that $\Omega_{A/\mathbf{F}_p}=0$. Is the Frobenius map on $A$ surjective?

Some context:

i. The converse is clearly true.
ii. The answer is yes if $A$ is a field, or of finite type.

Let $A$ be an $\mathbf{F}_p$-algebra such that $\Omega_{A/\mathbf{F}_p}=0$. Is the Frobenius map on $A$ surjective?

Some context:

i. The converse is clearly true.
ii. The answer is yes if $A$ is a field, or of finite type (the latter with a somewhat silly interpretation).