No. As described above, most stars end up as brown dwarfs, white dwarfs, or neutron stars, which are ejected from their galaxies. A body like a brown dwarf will gradually lose its atoms to the interstellar medium. Due to some counterintuitive thermodynamics, theythese atoms are probably eventually spontaneously ionized (Baez 2004). Baez gives a general argument that takes into account the cosmological environment, but to get the basic idea, I like the following argument given by Peierls 1979. Take a gas of hydrogen atoms. The sum $Z=\Sigma_{n=0}^\infty e^{-\beta E_n}$ diverges, so in the limit of low concentration, where $n$ can go arbitrarily high, the probability of any discrete state is zero.) Although the temperature is also going down over time, it reaches a finite limit, which is set by the Hawking radiation associated with the cosmological horizon.