To get a texture on your reflections, you must use _Object Coordinates._

* For whatever reason, you also have to select the light ("Light2" in my case) for the Texture Coordinate node so that the light's transforms are transferred to the coordinates in the node tree (usually it's automatic).<br>
[![enter image description here][1]][1]

* If you want to keep the coordinate values in $[0,1]$, make sure that your lamp is set to either Square or Disk with a Size of 1m, and in the node tree, add `0.5` to the $x$ and $y$.<br>
[![enter image description here][2]][2]<br>
(You can still make the light with any aspect ratio by changing the object scale.)

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Examples
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[![enter image description here][3]][3]<br>
[![enter image description here][4]][4]<br>
[![enter image description here][5]][5]<br>
For the last example, you can reduce the noise and render time by setting the shape to Disk so that the black corners aren't being sampled.

  [1]: https://i.sstatic.net/XNXII.png
  [2]: https://i.sstatic.net/9S5gP.png
  [3]: https://i.sstatic.net/6GcLR.png
  [4]: https://i.sstatic.net/I9nh1.png
  [5]: https://i.sstatic.net/iWcu8.png