This may seem like a weird question, but something got me thinking about it just recently.
The Sun's core is composed of mainly hydrogen and helium, and is present in the form of a extremely hot supercrushed plasma. The Sun's core is mind-bogglingly dense, about 150,000 kg/m3 (about 15x denser than lead, 7x denser than uranium, 6x denser than osmium). The density can get extremely high at the center of stars. This leads me to think that the solar core, due to the immense amount of atoms packed together would behave like a extremely hard solid, as per my understanding, most dense metals (excluding gold) are extremely hard, like tungsten.
I decided to dig into it a bit on the Internet, but whatever information I got were merely about the pressure at the center of the Sun, and not about the hardness of the solar core. By hardness, I mean having stiffness/rigidity, an ability to retain a certain shape when subjected to anisotropic stress.
To clarify things a bit: Supposing we submerged an "indestructible" observer really deep into the Sun, just inside the solar core. Supposing we got the indestructible observer to throw a punch randomly inside the Sun, what would this observer feel? More specifically, would the observer perceive the solar core material as being extremely hard, like a solid, or would it act like an extremely viscous fluid?
TL;DR
Would the solar core be extremely stiff and hard? Or would it simply behave like a dense and viscous gas?

The solar core can be considered soft in a relative sense (compared to other materials at the same density), but hard and incompressible in an absolute sense. The material behaves almost exactly like a perfect gas but would be as viscous as ketchup.
The equation of state is that of a perfect monatomic gas and thus the pressure $P \propto \rho^{\alpha}$, with $\alpha \sim 4/3$ in the solar core, where heat transport is dominated by radiative diffusion. This is a "soft equation of state" - the material is highly compressible - it takes a small fractional increase in pressure to produce a compression. For most solids, $\alpha$ would be in double figures and they are approximately incompressible.
However, what you are asking about could be represented by the "bulk modulus" (Young's modulus and Shear modulus are not meaningfully defined for a fluid). This is roughly equal to the pressure of a gas and is a measure of how much force in an absolute sense is required to change the volume of something. At the centre of the Sun, this is $2\times 10^{16}$ Pa. This can be compared with the bulk modulus of diamond which is $4\times 10^{11}$ Pa. Thus in that sense, the solar core is much harder than a diamond to compress.
In terms of viscosity, the microscopic kinematic viscosity in the solar core is of order $10^{-4}$ m$^2$/s (Ruediger & Kitchatinov 1996) and hence a dynamic viscosity of 15 Pa s. For comparison, the kinematic viscosity of water at 293K is $10^{-6}$ m$^2$/s and the dynamic viscosity is  $10^{-3}$ Pa s. Thus the solar interior fluid is 100-10000 times more viscous than water, depending on how viscosity is defined. Fluids of comparable viscosity would be honey or ketchup. The viscosity of solids (like rocks) meanwhile, is of order $\sim 10^{20}$ Pa s.
To understand why the Sun behaves like a perfect gas, one must compare the interaction energies between the particles with their kinetic energies. At a density of 150000 kg/m$^3$ the mean separation of protons and electrons is $\sim 2\times 10^{-11}$ m, with a mutual Coulomb energy of about 100 eV. The kinetic energy of the particles is $3k_BT/2$, and with temperature $T \sim 1.5\times 10^7$ K in the solar core, this is about 1000 eV. Thus the kinetic energies are much greater than the Coulombic binding energies and so the particles behave like a gas. To "freeze" into a solid you would need the binding energy to be about 100 times the kinetic energy, which would require much higher densities at that temperature.
