This result holds less obviously for Brownian motion with constant drift, not just $0$ drift. It is critical that the starting point is centered on the interval and it fails otherwise.

[Stern, F. An Independence in Brownian Motion with Constant Drift. *The Annals of Probability*, Vol. 5 (1977), 571-572.][1]

This holds for biased random walks because reflecting the paths to one boundary point gives paths to the other boundary with a constant magnification of probability. Taking the limit shows that the same is true for Brownian motion with constant drift. 


  [1]: https://projecteuclid.org/journals/annals-of-probability/volume-5/issue-4/An-Independence-in-Brownian-Motion-with-Constant-Drift/10.1214/aop/1176995763.full