  We compute exactly the statistics of the number of records in a discrete-time
random walk model on a line where the walker stays at a given position with a
nonzero probability $0\leq p \leq 1$, while with the complementary probability
$1-p$, it jumps to a new position with a jump length drawn from a continuous
and symmetric distribution $f_0(\eta)$. We show that, for arbitrary $p$, the
statistics of records up to step $N$ is completely universal, i.e., independent
of $f_0(\eta)$ for any $N$. We also compute the connected two-time correlation
function $C_p(m_1, m_2)$ of the record-breaking events at times $m_1$ and $m_2$
and show it is also universal for all $p$. Moreover, we demonstrate that
$C_p(m_1, m_2)< C_0(m_1, m_2)$ for all $p$, indicating that a nonzero $p$
induces additional anti-correlations between record events. We further show
that these anti-correlations lead to a drastic reduction in the fluctuations of
the record numbers with increasing $p$. This is manifest in the Fano factor,
i.e. the ratio of the variance and the mean of the record number, which we
compute explicitly. We also show that an interesting scaling limit emerges when
$p \to 1$, $N \to \infty$ with the product $t = (1-p)\, N$ fixed. We compute
exactly the associated universal scaling functions for the mean, variance and
the Fano factor of the number of records in this scaling limit.
