GDP per Capita in Panel Data: Must the base year be the same for all cross-sections?
Base rates account for the change in value of a currency that happens overtime because of inflation.
Not only do the base rates need to be the same, but if you are including all of these in the same panel, then the currencies need to be in the same unit as well, otherwise, the different country GDP's are not comparable.
Fortunately, most of the big international databases with GDP time series usually have multiple versions, with different base years, and converted to common currencies (usually USD) that you can use to assemble what you need.  Just a few places to start you off:
FRED
World Bank WDI
OECD Data
Note: for GDP data, these sites often present data from the same sources