  We propose a stellar model under the $f(\mathbb{T},\mathcal{T})$ gravity
following Mazur-Mottola's conjecture [Mazur (2001), Mazur (2004)] known as
gravastar which is generally believed as a viable alternative to black hole.
The gravastar consists of three regions, viz., (I) Interior region, (II)
Intermediate shell region, and (III) Exterior region. The pressure within the
interior core region is assumed to be equal to the constant negative
matter-energy density which provides a constant repulsive force over the thin
shell region. The shell is assumed to be made up of fluid of ultrarelativistic
plasma and following the Zel'dovich's conjecture of stiff fluid [Zeldovich
(1972)] it is also assumed that the pressure which is directly proportional to
the matter-energy density according to Zel'dovich's conjecture, does cancel the
repulsive force exerted by the interior region. The exterior region is
completely vacuum and it can be described by the Schwarzschild solution. Under
all these specifications we find out a set of exact and singularity-free
solutions of the gravastar presenting several physically valid features within
the framework of alternative gravity, namely $f(\mathbb{T},\mathcal{T})$
gravity [Harko (2014)], where the part of the gravitational Lagrangian in the
corresponding action is taken as an arbitrary function of torsion scalar
$\mathbb{T}$ and the trace of the energy-momentum tensor $\mathcal{T}$.
