Ortiz is actually an example of a player who did seem to elevate his game in key situations. Baseball Reference has a stat dealing with leverage (high, medium and low). They define leverage as how a situation affects win probability. Man on second late in a tie game = high leverage, same situation in a blowout = low leverage. Somewhat subjective, but instructional:
Ortiz's stats by leverage:
High - 1998 PAs, .292/.388/.556/.943
Medium - 3921 PAs, .299/.388/.571/.959
Low - 4162 PAs, .271/.368/.532/.900
His career slashline was .286/.380/.552/.931
He did seem to "bear down" when it counted. But the numbers still are not all that different than his overall career.