  We present several classes of solitons in ($1+1$)-dimensional models where
the standard canonical kinetic term is replaced by a Dirac-Born-Infeld (DBI)
one. These are static solutions with finite energy and different properties,
namely, they can have compact support, or be kink or lump-like, according to
the type of potential chosen, which depend on the DBI parameter $\beta$.
Through a combination of numerical and analytical arguments, by which the
equation of motion is seen as that corresponding to \emph{another} canonical
model with a new $\beta$-dependent potential and a $\beta$-deformed energy
density, we construct models in which increasing smoothly the DBI parameter
both the compacton radius, the thickness of the kink and the width of the lump
get modified until each soliton reaches its standard canonical shape as $\beta
\rightarrow \infty$. In addition we present compacton solutions whose canonical
counterparts are not compact.
