No it isn't, but I had to dig quite deep to get a counterexample. Let us look at smooth $(D^7, \partial D^7)$-bundles over $S^2$, i.e $D^7 \to E \overset{\pi} \to S^2$ with an identification $\partial E \cong S^2 \times \partial D^7$. These are classified by a map
$$f : S^2 \to BDiff_\partial(D^7)$$
to the classifying space of the group of diffeomorphisms of $D^7$ relative to the boundary. I will show that there exists an $f$ which does not extend to $\mathbb{CP}^2$.

Smoothing theory gives an identification
$$BDiff_\partial(D^7) \simeq \Omega^7\left(\frac{TOP(7)}{O(7)}\right)$$
and by page 246 of Kirby--Siebenmann the map
$$\Omega^7\left(\frac{TOP(7)}{O(7)}\right) \to \Omega^7\left(\frac{TOP}{O}\right)$$
is 2-connected, so there is a surjection
$$\pi_2(BDiff_\partial(D^7)) \to \pi_9(\tfrac{TOP}{O}) \quad \quad(*).$$
To find an example of such a bundle which does not extend over $\mathbb{CP}^2$, it is therefore enough to show that the map
$$- \circ \eta : \pi_9(\tfrac{TOP}{O}) \to \pi_{10}(\tfrac{TOP}{O}) \quad \quad(**)$$
given by precomposition with the (suitably suspended) Hopf map is not trivial.

By surgery theory $\pi_n(\tfrac{TOP}{O}) \cong \Theta_n$, the group of homotopy $n$-spheres, and a homotopy $n$-sphere has a unique Spin structure, determining a $\pi_n(\tfrac{TOP}{O}) \cong \Theta_n \to \Omega^{Spin}_n$, which we can further compose with the Atiyah--Bott--Shapiro map
$$\alpha : \Omega^{Spin}_n \to KO_n.$$
Applying this construction to both sides, the map $(**)$ above becomes the map
$$- \circ \eta : KO_9 = \mathbb{Z}/2 \to KO_{10} = \mathbb{Z}/2,$$
which is an isomorphism. Finally, it is well-known that
$$\pi_n(\tfrac{TOP}{O}) \cong \Theta_n \to \Omega^{Spin}_n \to KO_n$$
is surjective for $n \equiv 1,2 \mod 8$.

So choose a $f' \in \pi_9(\tfrac{TOP}{O}) \cong \Theta_9$ which is non-trivial under the Atiyah--Bott--Shapiro map, and use that $(*)$ is surjective to lift it to a $f \in \pi_2(BDiff_\partial(D^7))$. This has the desired property.

**Addendum**

The following gambit can be used to get a closed fibre example. The product of the natural maps $SO(n+1) \to Diff^+(S^n)$ and $Diff_\partial(D^n) \to Diff^+(S^n)$ gives a map
$$SO(n+1) \times Diff_\partial(D^n) \to Diff^+(S^n)$$
which is a weak homotopy equivalence (though not a homomorphism). Thus the map
$$BDiff_\partial(D^7) \to BDiff^+(S^7)$$
is (split) injective on all homotopy groups, so the example above also gives an oriented $S^7$-bundle over $S^2$ which cannot extend to $\mathbb{CP}^2$.