You could create your own SliverAppBar or you can divide them in 2 items, a SliverAppBar and a SliverPersistentHeader
class Home extends StatefulWidget {
@override
_HomeState createState() => _HomeState();
}
class _HomeState extends State<Home>
with SingleTickerProviderStateMixin {
TabController controller;
TextEditingController textController = TextEditingController();
@override
void initState() {
super.initState();
controller = TabController(
length: 3,
vsync: this,
);
}
@override
void dispose(){
super.dispose();
controller.dispose();
textController.dispose();
}
@override
Widget build(BuildContext context) {
return Scaffold(
body: CustomScrollView(
slivers: <Widget>[
SliverAppBar(
leading: const Icon(Icons.menu),
title: TextField(
controller: textController,
textInputAction: TextInputAction.search,
decoration: InputDecoration(
isDense: true,
hintText: 'Search Bar',
hintStyle: TextStyle(color: Colors.black.withOpacity(.5), fontSize: 16),
border: InputBorder.none
)
),
snap: true,
floating: true,
actions: [
IconButton(
icon: const Icon(Icons.search),
onPressed: () => print('searching for: ${textController.text}'),
)
]
),
//This is Where you create the row and your tabBar
SliverPersistentHeader(
delegate: MyHeader(
top: Row(
children: [
for(int i = 0; i < 4; i++)
Expanded(
child: OutlineButton(
child: Text('button $i'),
onPressed: () => print('button $i pressed'),
)
)
]
),
bottom: TabBar(
indicatorColor: Colors.white,
tabs: [
Tab(text: 'Tab 1'),
Tab(text: 'Tab 2'),
Tab(text: 'Tab 3'),
],
controller: controller,
),
),
pinned: true,
),
SliverFillRemaining(
child: TabBarView(
controller: controller,
children: <Widget>[
Center(child: Text("Tab one")),
Center(child: Text("Tab two")),
Center(child: Text("Tab three")),
],
),
),
],
),
);
}
}
//Your class should extend SliverPersistentHeaderDelegate to use
class MyHeader extends SliverPersistentHeaderDelegate {
final TabBar bottom;
final Widget top;
MyHeader({this.bottom, this.top});
@override
Widget build(BuildContext context, double shrinkOffset, bool overlapsContent) {
return Container(
color: Theme.of(context).accentColor,
height: math.max(minExtent, maxExtent - shrinkOffset),
child: Column(
mainAxisSize: MainAxisSize.min,
children: [
if(top != null)
SizedBox(
height: kToolbarHeight,
child: top
),
if(bottom != null)
bottom
]
)
);
}
/*
kToolbarHeight = 56.0, you override the max and min extent with the height of a
normal toolBar plus the height of the tabBar.preferredSize
so you can fit your row and your tabBar, you give them the same value so it
shouldn't shrink when scrolling
*/
@override
double get maxExtent => kToolbarHeight + bottom.preferredSize.height;
@override
double get minExtent => kToolbarHeight + bottom.preferredSize.height;
@override
bool shouldRebuild(SliverPersistentHeaderDelegate oldDelegate) => false;
}

UPDATE
A NestedScollView let you have 2 ScrollViews so you can control the inner scroll with the outer (just like you want with a TabBar)
class HomePage extends StatefulWidget {
@override
_HomePageState createState() => _HomePageState();
}
class _HomePageState extends State<HomePage> {
TextEditingController textController = TextEditingController();
List<String> _tabs = ['Tab 1', 'Tab 2', 'Tab 3'];
// Your tabs, or you can ignore this and build your list
// on TabBar and the TabView like my previous example.
// I don't create a TabController now because I wrap the whole widget with a DefaultTabController
@override
void initState() {
super.initState();
}
@override
void dispose() {
super.dispose();
textController.dispose();
}
@override
Widget build(BuildContext context) {
return Scaffold(
body: DefaultTabController(
length: _tabs.length, // This is the number of tabs.
child: NestedScrollView(
headerSliverBuilder: (BuildContext context, bool innerBoxIsScrolled){
return <Widget>[
SliverOverlapAbsorber(
handle: NestedScrollView.sliverOverlapAbsorberHandleFor(context),
sliver: SliverAppBar(
elevation: 0.0,
leading: const Icon(Icons.menu),
title: TextField(
controller: textController,
textInputAction: TextInputAction.search,
decoration: InputDecoration(
isDense: true,
hintText: 'Search Bar',
hintStyle: TextStyle(
color: Colors.black.withOpacity(.5),
fontSize: 16),
border: InputBorder.none)
),
snap: true,
floating: true,
actions: [
IconButton(
icon: const Icon(Icons.search),
onPressed: () => print('searching for: ${textController.text}'),
)
]
),
),
SliverPersistentHeader(
delegate: MyHeader(
top: Row(children: [
for (int i = 0; i < 4; i++)
Expanded(
child: OutlineButton(
child: Text('button $i'),
onPressed: () => print('button $i pressed'),
))
]),
bottom: TabBar(
indicatorColor: Colors.white,
tabs: _tabs.map((String name) => Tab(text: name)).toList(),
),
),
pinned: true,
),
];
},
body: TabBarView(
children: _tabs.map((String name) {
return SafeArea(
child: Builder(
// This Builder is needed to provide a BuildContext that is
// "inside" the NestedScrollView, so that
// sliverOverlapAbsorberHandleFor() can find the
// NestedScrollView.
// You can ignore it if you're going to build your
// widgets in another Stateless/Stateful class.
builder: (BuildContext context) {
return CustomScrollView(
// The "controller" and "primary" members should be left
// unset, so that the NestedScrollView can control this
// inner scroll view.
// If the "controller" property is set, then this scroll
// view will not be associated with the NestedScrollView.
// The PageStorageKey should be unique to this ScrollView;
// it allows the list to remember its scroll position when
// the tab view is not on the screen.
key: PageStorageKey<String>(name),
slivers: <Widget>[
SliverOverlapInjector(
// This is the flip side of the SliverOverlapAbsorber
// above.
handle: NestedScrollView.sliverOverlapAbsorberHandleFor(context),
),
SliverPadding(
padding: const EdgeInsets.all(8.0),
sliver: SliverFixedExtentList(
itemExtent: 48.0,
delegate: SliverChildBuilderDelegate(
(BuildContext context, int index) {
return ListTile(
title: Text('Item $index'),
onTap: () => print('$name at index $index'),
);
},
childCount: 30,
),
),
),
],
);
},
),
);
}).toList(),
),
),
));
}
}