This is the scene:
The three points $A$, $B$, $C$ and the unit spheres around.
Then the plane through the three points.
It is clear that the two parallel planes in distance $\pm 1$ to that plane are among the sought ones.
I would agree with David that this is mostly a decision per sphere, if the plane touches above or below (relative to the containing plane).
Which seems to lead to eight choices.
We can describe an arbitray plane as
n \cdot x = d
where $n$ is a unit normal vector of the plane and $d$ is the (signed) distance of the plane to the origin.
The spheres are described by
(x - P)^2 = 1
The vectors $x$ within the plane must not enter the spheres:
(x - P)^2 \ge 1
At three points the plane touches the spheres:
n \cdot x_P = d \quad\quad
(x_P - P)^2 = 1 \\
So we have the system
E: n \cdot x = d \\
(x - A)^2 \ge 1 \\
(x - B)^2 \ge 1 \\
(x - C)^2 \ge 1 \\
n \cdot x_A = d \quad\quad
(x_A - A)^2 = 1 \\
n \cdot x_B = d \quad\quad
(x_B - B)^2 = 1 \\
n \cdot x_C = d \quad\quad
(x_C - C)^2 = 1
for all $x \in E$ and $13$ real unknowns $n = (n_1, n_2, n_3)^\top$, $d$ and $x_A$, $x_B$, $x_C$.