To address the problem, we need to determine whether every finite obtuse-angled set can indeed be extended to an infinite obtuse-angled set.

### Definitions and Assumptions

1. Obtuse-Angled Set: By definition, a set of points is described as obtuse-angled if no three points are collinear and every triangle formed by any three points from the set has at least one angle greater than $91^\circ$.

The problem requires showing that for any finite obtuse-angled set, we can keep adding points indefinitely without losing the obtuse-angled property.

### Consideration of the Triangle's Angles

- Assume we have a finite obtuse-angled set $S$. Each triangle formed by the points of $S$ has at least one angle that is strictly greater than $91^\circ$. This implies that the remaining angles of each triangle are acute.

- If we add a point $P$ to extend $S$, then for triangles involving $P$ and two other points from $S$, we must ensure at least one angle remains greater than $91^\circ$.

### Adding Points Indefinitely

- To extend $S$ to an infinite obtuse-angled set, we strategically add points such that the obtuse-angled property is preserved. This can be done by placing each new point sufficiently far such that new triangles formed have their largest angle opposite the largest side, satisfying the obtuse condition.

- Alternatively, each new point can be added under consideration of spherical geometry or by maintaining a specific geometric arrangement, such as spreading points out on a curve. This ensures new triangles will have obtuse angles due to specific distances or geometric configurations, maintaining non-collinearity and obtuseness.

### Conclusion

The idea is to consistently add points in positions or distances that do not permit the formation of purely acute or right triangles, thus maintaining at least one obtuse angle in each new triangle.

Hence, the reference answer is affirmative:
$$\boxed{\text{Yes}}$$
This indicates that every finite obtuse-angled set can indeed be extended to an infinite obtuse-angled set.