  We study an extension of the generalized excited random walk (GERW) on
$\mathbb{Z}^d$ introduced in [Ann. Probab. 40 (5), 2012, [7]] by Menshikov,
Popov, Ram\'irez and Vachkovskaia. Our extension consists in studying a version
of the GERW where excitation depends on a random environment. Given $p \in
(0,1]$ (a parameter of the model) whenever the process visits a site for the
first time, with probability $p$ it gains a drift in a given direction (could
be any direction of the unit sphere). Otherwise, with probability $1-p$, it
behaves as a $d$-martingale with zero-mean vector. Whenever the process visits
an already-visited site, the process acts again as a $d$-martingale with
zero-mean vector. We refer to the model as a GERW in Bernoulli environment, in
short $p$-GERW. Under the same hypothesis of [7] (bounded jumps, uniform
ellipticity), we show that the $p$-GERW is ballistic for all $p\in (0,1]$.
Under the stronger assumptions that the increments of the regeneration times
associated to the $p$-GERW are i.i.d. (condition which is satisfied, for
example, for the excited random walk in a Bernoulli i.i.d. environment), we
also obtain a Law of Large Numbers and a Central Limit Theorem.
