Faça de entrada reactjs de posição fixa

<div style={{display: 'flex', flexDirection: 'row', alignItems: 'center', justifyContent: 'center' }}>

    <h1>Do Not Move Me</h1>

    <div style={{ position: 'relative', marginLeft: 20 }}>

        <input
        placeholder='Enter Name'
        onChange={this._displayButton}
        value={this.state.nameText}
        />
        {this.state.textEntered ?
            <button type="submit" style={{ position: 'absolute', top: '-10px', right: '-10px', borderRadius: '50%', height: 40, width: 40, backgroundColor: '#881587', border: 'none' }} /> : null
        }

    </div>

</div>