Knowing the answer to this question would help me answer the following question:
$A$ is an $m\times n$ matrix with $m>n$, and let $A=\hat{Q}\hat{R}$ be a reduced QR factorization. Suppose $\hat{R}$ has $k$ nonzero diagonal entries for some $k$ with $0\leq k<n$, what does this imply about the rank of $A$? Exactly $k$? At least $k$? At most $k$?
Answering the title question: No.
The rank of a matrix equals its column rank, which by definition is the number of linearly independent columns. Adding one column will increase the column rank by one if it is independent from the others and will leave the column rank unchanged if it is dependent on the others.