Pebble game lower bound? This paper says pebble games have super linear lower bound for every fixed $k$ https://dl.acm.org/citation.cfm?doid=62.322433.

Why is it not considered proof of constructive example for a function in $NP$ which requires superlinear runtime?
The time hierarchy theorem gives, for every $k$, a function in P with a runtime lower bound of $\Omega(n^k)$. Unfortunately, this is not enough to separate P from NP: to do this, we need a function in NP with a superpolynomial runtime lower bound.

One popular approach for tackling the P vs NP question is via circuits. The best lower bounds for explicit functions are only linear. Perhaps this is the context in which you heard of superlinear lower bounds as a seeming barrier.