Well, the data of map points on the surface of the Earth lies in a 2 dimensional manifold.  Even though the surface is very irregular (mountains, etc.).  And this surface exists in a 3 dimensional space.

While I have a little experience with manifolds, the second part of your question about finding a lower dimensional subspace (I assume you mean manifold) suggested to me you may be asking:  is there any set of data points in a higher dimensional space that cannot be represented as a lower dimensional manifold?

If one takes literally the examples at https://en.wikipedia.org/wiki/Manifold then no.  Look at the example where they have multiple circles or line segments that constitute a single manifold with disconnected parts.  There is no limit on how small the circles can be, they could be points, and there is no limit on the number of them, so all the points in a higher dimensional space could be mapped one for one to such a manifold.

One can argue then it is really something different, but the definition appears to have excess flexibility.

The article begins by discussing how the manifold approaches Euclidean space about a point.  Since no limit for how small the space must be is imposed, this forms no restriction.

There are a class of manifolds which must be differentiable.  This might impose some limitations.  An expert in fractal geometry would be needed to figure it out, I could not do it.  For example, such a space could be folded (gently, to keep it differentiable) and wrapped around to nearly fill a higher dimensional space.  But could it touch every point?  Maybe not.  But if the higher dimensional space is discretized, however crudely, even to the extent of having a Planck Length below which it cannot go, then a lower dimensional differentiable manifold could touch every point of it.

for example refer the attached file