No I wouldn't. Fortunately, that's what the Internet---the One and Only---has proxy servers for. Google's proxy server on Ganymede would cache the 1,000,000,000 most frequently used searches from the rest of the solar system, so you typically wouldn't have
to wait more than a few milliseconds for your response. The connection between Google's proxy server on Ganymede and that on Earth would probably still be part of the Internet. "Probably", because it could
be part of Google's intranet instead. Notice the lowercase in "intranet"---there will be many intranets in the solar system, just as there are many intranets on Earth now.