We had a similar feeling in Jamaica, and now in the Cayman it’s exactly the same, animals just wander freely around the islands and can show up anywhere. Ok, in South of Europe we have cats and dogs, in the north foxes and squirrels, but in the Caymans we have lizards, chickens, land crabs and we will see what more. Yes, land crabs, we did not know they existed but crabs have evolved to be totally independent from water. Not totally, they still need to spawn in the water but beside that they live inland taking the natural places of normal land predators like rabbits.
The blue Iguana is a special type of lizard that only exists in the Caymans Islands and is in danger of extinction so they are quite protected and difficult to spot, they are easier to be seen in the botanic garden of the island.
There also hundreds of birds around, but the Islands have one called by the people chin chin due to the noise they make, that are so used to humans that there are stories of them scaring people away. They are the same type of plague birds as the doves, but dark in this case not white and more silent.