  In this paper, we define the quaternionic Fock spaces
$\mathfrak{F}_{\alpha}^p$ of entire slice hyperholomorphic functions in a
quaternionic unit ball $\mathbb{B}$ in $\mathbb{H}.$ We also study growth
estimate and various results of entire slice regular functions in these spaces.
The work of this paper is motivated by the recent work of \cite{alpa12} and
\cite{marco}.
