  In this paper, we prove that there exists a universal constant $C$, depending
only on positive integers $n\geq 3$ and $p\leq n-1$, such that if $M^n$ is a
compact free boundary submanifold of dimension $n$ immersed in the Euclidean
unit ball $\mathbb{B}^{n+k}$ whose size of the traceless second fundamental
form is less than $C$, then the $p$th cohomology group of $M^n$ vanishes. Also,
employing a different technique, we obtain a rigidity result for compact free
boundary surfaces minimally immersed in the unit ball $\mathbb{B}^{2+k}$.
