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There is a code for the Mount Everest

data = GeoElevationData[
GeoBoundingBox[
GeoDisk[Entity["Mountain", "MountEverest"], Quantity[5, "Miles"]]]]

ListPlot3D[data, MeshFunctions -> {#3 &}, ColorFunction -> "Rainbow", 
Filling -> Bottom, FillingStyle -> Red, 
Lighting -> {{"Ambient", White}}, BoundaryStyle -> None, 
Boxed -> False]

enter image description here

it is possible to do same thing for another mountain

for instance: mountain in bakhmaro

enter image description here

but unfortunately this isn't entity object in Mathematica.

Does there exist any way to do this?

Is it possible to import information from google earth about this mountain to construct data like Mount Everest?

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GeoElevationData has elevation data for the whole world. If you know the position (I hope I interpreted correctly the Wikipedia data from your link):

In[]:= p = GeoPosition[{FromDMS[{42, 19, 32.}], FromDMS[{41, 51, 06.}]}]
Out[]= GeoPosition[{42.3256, 41.8517}]

then you can evaluate your same commands, replacing the entity by that position:

data = GeoElevationData[GeoBoundingBox[GeoDisk[p, Quantity[5, "Miles"]]]]

ListPlot3D[Reverse[data], MeshFunctions -> {#3 &}, ColorFunction -> "Rainbow", Filling -> Bottom,
           FillingStyle -> Red, Lighting -> {{"Ambient", White}}, BoundaryStyle -> None,
           Boxed -> False, PlotRange -> All]
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