There may be some small local leagues and I know that some universities have club level (intra-campus, not inter-university) competitions, but generally the answer is "no".
We do like our baseball very much, but we usually consider a 5-hour long game to be too long - I cannot possibly imagine trying to keep up with a game that could span days. Maybe it has something to do with our drinking beer while watching baseball - though I've personally tried a few times, it doesn't do my body well to be drinking for 3 or 4 days straight! I don't know how any of your die-hard cricket fans can hold a steady job?!

Good luck with your cricket matches !
Also, so sorry to hear of all the damage and deaths from the drought and fires in Australia? Is there any end in sight to the hot, dry weather for you? Maybe that will also
keep the cricket matches shorter - only mad dogs and Englishmen go out in the mid-day sun, right ?