Answer
False.
Work Step by Step
If there is an asymptote at $y=0$, there doesn't exist a real value of $x$ such that $y=0$. Using $y=0$ with $y=\ln x$ gives $$0 = \ln x$$ However, $\ln (1)=0$. So, $x=1$ is a value at $y=0$. This means that there cannot be an asymptote at $y=0$.
