There's a paper of H. L. Royden

"Holomorphic fiber bundles with hyperbolic fiber", Proc. AMS, Volume 43, Number 2, April 1974

which proves

"Theorem: The holomorphic fiber bundles with hyperbolic fiber
M and base B are in natural one-to-one correspondence with the homomor-
phisms of the fundamental group of B into the group of biholomorphic
automorphisms of M onto itself. "

In other words, it appears that holomorphic disc bundles are naturally flat.

**Note**: the first projection $p_1: \mathbb{B}^2\to \mathbb{D}$ of the real 4-ball to the disc is not a holomorphic bundle of discs, because it is not locally trivial. To see why, suppose we have an isomorphism $f:p_1^{-1}(D)\to D\times \mathbb{D}$ of bundles over a sufficently small disc $D$ of radius $r<1$ around zero. Write $f(x,y) = (x, f_2(x,y))$.

Now choose $y_0\in p_1^{-1}(0)$ with $\sqrt{1-r^2} <|y_0|< 1$, so that it is sent via the second projection $p_2:\mathbb{B}^2\to\mathbb{D}$ outside the disc of radius $\sqrt{1-r^2}$.

By assumption, $p_1$ has a section $s$ over $D$ passing through $y_0$ -- this is just a level set of $f_2$. Composing, we obtain $ h=p_2\circ s : D\to \mathbb{D}$. I claim $h$ fails the maximum modulus principle for holomorphic functions. Why? Any circle of sufficiently large radius in $D$ must be sent by $h$ inside a disc of radius strictly less than $|y_0|$ around $0$. Meanwhile, the center of the disc $D$ is sent to $p_2(y_0)$, which has norm equal to $|y_0|$. This goes against the maximum principle or the Gauss mean value theorem, if you like.