What is Brane Inflation and how does it describe inflation process? Can it be eternal?
Brane inflation is, roughly speaking, a paradigm that propose an identification of brane/anti-brane annihilation processes with some types of cosmological inflation (typically hybrid ones).
Your question about the possibility of producing an eternal inflation scenario in the "brane inflation" paradigm cannot be answered unless you ask for a particular scenario within this paradigm.Despite of the later, I think is safe to say that in basic scenarios eternal inflation is not possible. The minimum of the potential for a tachyon field in a $D3$/ anti-$D3$-brane annihilation (in flat space) is reached at finite distances in moduli space, in other words, a stable vacuum for the pair is reached in finite time (see the review I attach).
In String Cosmology, Brane inflation is a mathematical realisation of the Cosmic Inflation in the very early universe within the brane world framework. It usually depends on how the brane inflationary universe originated, eternal inflation of the false vacuum type is too possible. Eternal inflation of the false vacuum type inevitably happens in this case due to the tunneling process.