We introduce a $1+1$-dimensional temperature-dependent model such that the
classical ballistic deposition model is recovered as its zero-temperature
limit. Its $\infty$-temperature version, which we refer to as the $0$-Ballistic
Deposition ($0$-BD) model, is a randomly evolving interface which, surprisingly
enough, does {\it not} belong to either the Edwards--Wilkinson (EW) or the
Kardar--Parisi--Zhang (KPZ) universality class. We show that $0$-BD has a
scaling limit, a new stochastic process that we call {\it Brownian Castle} (BC)
which, although it is "free", is distinct from EW and, like any other
renormalisation fixed point, is scale-invariant, in this case under the $1:1:2$
scaling (as opposed to $1:2:3$ for KPZ and $1:2:4$ for EW). In the present
article, we not only derive its finite-dimensional distributions, but also
provide a "global" construction of the Brownian Castle which has the advantage
of highlighting the fact that it admits backward characteristics given by the
(backward) Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related
Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar,
Ann. Probab., '04]). Among others, this characterisation enables us to
establish fine pathwise properties of BC and to relate these to special points
of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller
process on a suitable space of c\`adl\`ag functions and determine its long-time
behaviour. At last, we give a glimpse to its universality by proving the
convergence of $0$-BD to BC in a rather strong sense.