Can an AngularJS controller inherit from another controller in the same module?

Yes, it can but you have to use the $controller service to instantiate the controller instead:-

var app = angular.module('angularjs-starter', []);

app.controller('ParentCtrl', function($scope) {
  // I'm the sibling, but want to act as parent
});

app.controller('ChildCtrl', function($scope, $controller) {
  $controller('ParentCtrl', {$scope: $scope}); //This works
});

In case you are using vm controller syntax, here is my solution:

.controller("BaseGenericCtrl", function ($scope) {

    var vm = this;
    vm.reload = reload;
    vm.items = [];

    function reload() {
        // this function will come from child controller scope - RESTDataService.getItemsA
        this.getItems();
    }
})

.controller("ChildCtrl", function ($scope, $controller, RESTDataService) {
    var vm = this;
    vm.getItems = RESTDataService.getItemsA;
    angular.extend(vm, $controller('BaseGenericCtrl', {$scope: $scope}));
})

Unfortunately, you can't use $controller.call(vm, 'BaseGenericCtrl'...), to pass current context into closure (for reload()) function, hence only one solution is to use this inside inherited function in order to dynamically change context.