  Given a Banach space $\mathcal X$, let $x$ be a point in
$\text{ball}(\mathcal X)$, the closed unit ball of $\mathcal X$. One says that
$x$ is a strongly extreme point of $\text{ball}(\mathcal X)$ if it has the
following property: for every $\varepsilon>0$ there is $\delta>0$ such that the
inequalities $\|x\pm y\|<1+\delta$ imply, for $y\in\mathcal X$, that
$\|y\|<\varepsilon$. We are concerned with certain subspaces of $H^\infty$, the
space of bounded holomorphic functions on the disk, that arise upon imposing
finitely many linear constraints and can be viewed as finite-dimensional
perturbations of $H^\infty$. It is well known that the strongly extreme points
of $\text{ball}(H^\infty)$ are precisely the inner functions, while the (usual)
extreme points of this ball are the unit-norm functions $f\in H^\infty$ with
$\log(1-|f|)$ non-integrable over the circle. Here we show that similar
characterizations remain valid for our perturbed $H^\infty$-type spaces. Also,
we investigate to what extent a non-inner function can differ from a strongly
extreme point.
