lots of points

Let our surface be a sphere with center $O$ and radius $R=\dfrac{12}{\pi }$ .

$AA'$ , $BB'$ , $CC'$ are three diameters that are perpendicular to each other.
$D$ , $E$ , $F$ are the midpoints of the minor arcs defined by two of the points $A$ , $B$ and $C$ as seen in the diagram. These minor arcs (geodesics) are segments of corresponding great circles, therefore their length represent the shortest distance of the two points they join.

We will prove that $A$ , $B$ , $C$ , $D$ , $E$ and $F$ can be the required points.

First, we calculate the distances of $A$ from the other five points.
Each of the arcs $\overset\frown{AB}$ , $\overset\frown{AE}$ , $\overset\frown{AC}$ , is subtended by a central angle $\varphi =\dfrac{\pi }{2}$ radians, hence the arc lengths are equal to $R\varphi =\dfrac{12}{\pi }\times \dfrac{\pi }{2}=6$ , i.e. the distance between $A$ and $B$ , $E$ , or $C$ is $\boxed{6}$ .
Since $D$ and $F$ are midpoints of $\overset\frown{AB}$ and $\overset\frown{AC}$ , the distance of $A$ from these two points is $\boxed{3}$ . Due to symmetry, all blue arcs have lengths of $3$ or $6$ .

Last thing is to show that the lengths of the magenta arcs are also integers. All three arcs are equal, so we go just for one of them.
For this, we use the Spherical Pythagorean Theorem on the right angled spherical triangle $BDE$ , $\left(\angle B=\dfrac{\pi }{2}\right)$ :
Let $\beta$ , $\epsilon$ , $\delta$ denote the lengths of the arcs $\overset\frown{DE}$ , $\overset\frown{BD}$ and $\overset\frown{BE}$ respectively.

Then, $\begin{aligned} & \cos \frac{\beta }{R}=\cos \frac{\varepsilon }{R}\cdot \cos \frac{\delta }{R} \\ & \Rightarrow \cos \frac{\beta }{R}={{\left( \cos \frac{3}{\frac{12}{\pi }} \right)}^{2}}={{\left( \cos \frac{\pi }{4} \right)}^{2}}=\frac{1}{2} \\ & \Rightarrow \frac{\beta }{R}={{\cos }^{-1}}\left( \frac{1}{2} \right)=\frac{\pi }{3} \\ & \Rightarrow \beta =\frac{\pi }{3}\times R=\frac{\pi }{3}\times \frac{12}{\pi } \\ & \Rightarrow \beta =\boxed{4} \\ \end{aligned}$

To conclude, the distance between any points is an integer, and no more than three points share a line (i.e. a great circle).