Is there a mountain-shaped iceberg?
Ice being less dense than water, icebergs float to the degree they displace water, with about 1/8 of their mass above the surface. Icebergs are also dynamic, melting from exposure to sunlight, warm air and water, eroding, and fracturing, and pushed by both wind and waves. (Notably the descriptions used in the Wikipedia article generally apply to the shape of that small portion of an iceberg visible above the surface, not the gross geometry of the whole berg.)
Geophysicist Henry Pollack explored the potential stable positions of icebergs in a Physics Today article titled &quot;Tip of the Iceberg&quot;:

When a totally submerged lower-density body is released, the buoyancy
force causes it to rise until it reaches a floating equilibrium. The
tip then rests above the surface and the root below it, with the mass
of each determined by the density contrast between the floating solid
and the surrounding fluid.

A floating object is only stable to the extent that its center of gravity is aligned above its center of buoyancy, Pollack writes, &quot;...ice cylinders floating in water will stabilize in only two orientations—with the cylindrical axis either perpendicular or parallel to the water surface;&quot; With an idealized ice-like cylinder 9/10 the density of water, the mass can't be stable if it's taller than it is wide.
The geometry won't be much different using an idealized cone shape -- a cylinder widening along its length -- for your mountain. It can only be short and squat,  and if it's at all unbalanced at any point in its life, it'll flip to a new stable equilibrium, on its side or upside-down.