No: three dots having the same distances to each other form an equilateral triangle, and there is no way to add a fourth to have the same distance to all three of those. It would have to be on all three of the circles in this image:
(So it certainly isn't possible to do 10 dots!)
In general, you need $n-1$ dimensions for $n$ dots with this property: these dots will form the vertices of a regular $k$-simplex.
