No, Java doesn't have that ability.
It does have System.nanoTime(), but that just gives an offset from some previously known time. So whilst you can't take the absolute number from this, you can use it to measure nanosecond (or higher) precision.
Note that the JavaDoc says that whilst this provides nanosecond precision, that doesn't mean nanosecond accuracy. So take some suitably large modulus of the return value.
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