Can a solid object float on a frictionless liquid? Assuming the solid is less dense than the liquid, will a solid object float on a frictionless liquid?
I can imagine that due to the pressure gradient the object will float, but I can also imagine that without friction there would be no upwards force on the object and it would sink.
What would happen?
 A: Yes, it will float.
Floating is based on buoyancy.
Which is not related to surface properties like friction of either the object or the fluid.
The pressure causing the force keeping the object to keep floating is the hydrostatic pressure, also unrelated to surface properties; So it's about the pressure gradient you mention in the question.

Interestingly, it may be possible to just try it:
A frictionless fluid is at least related to superfluidity.
Superfluidity is stronger as it also excludes 'inner friction', the - viscosity;
In general it shows some interesting effects. At least a large part of that is related to being frictionless.
Extremely cold liquid Helium can be superfluid. That could certainly be used to try the experiment - on floating.

That experiment would not be hard - but has some tricky aspects:
The handling of cyrogenic Helium is not trivial,
and  great care has to be taken to cool down the object to float before using it.
If the object to float is formed like a small boat to make it light, there may be some unexpected complications. Very, very unexpected.
But the hardest part is to not get distracted by the other properties of superfluid Helium - which are closely related to surfaces.
A: Assume it sinks. In this case it will displace a volume of liquid, which is the same as to say that it will switch places with a volume of liquid. Hence a heavier volume of liquid will go up, and lighter volume of solid will go down. Do you see the issue?
The issue is that this configuration does not minimize the energy of the system. So it will not sink.
