The user manual of geonames.org shows a mountain as one of its examples, so it does seem like their data includes mountains. It looks like mountains will have the following data field values:
name: [mountain name, eg 'Säntis']
code: Mountain (MT)
You can download the latest datadump of all geonames data here (this is a ...
You can get that data from the World Database on Protected Areas (WDPA) from UN Environment and the International Union for Conservation of Nature (IUCN). The geospatial data for India is available for download as Shapefile or CSV here.