  In this paper we define the $p$-adic framed braid group ${\mathcal
F}_{\infty,n}$, arising as the inverse limit of the modular framed braids and
we give topological generators for ${\mathcal F}_{\infty, n}$. We also give
geometric interpretations for the $p$-adic framed braids. We then construct a
$p$-adic Yokonuma-Hecke algebra ${\rm Y}_{\infty,n}(u)$ as the inverse limit of
a family of classical Yokonuma-Hecke algebras. These are quotients of the
modular framed braid groups over a quadratic relation. We also give topological
generators for ${\rm Y}_{\infty,n}(u)$. Finally, we construct on this new
algebra a linear trace that supports the Markov property.
