CachedImage.js

import React from 'react';
import { Image, View, StyleSheet } from 'react-native';
import { Asset, FileSystem } from 'expo';
import sha256 from 'crypto-js/sha256';

export default class CachedImage extends React.Component {
  state = {
    source: null,
  };

  async componentDidMount() {
    let source = this.props.source;

    try {
      if (typeof source === 'number') {
        await Asset.fromModule(source).downloadAsync();
      } else if (source && source.uri) {
        let parts = source.uri.split('.');
        let ext = parts[parts.length - 1];
        let name = sha256(source.uri);
        let filepath = `${FileSystem.documentDirectory}${name}.${ext}'`;
        let { exists } = await FileSystem.getInfoAsync(filepath);
        if (exists) {
          source = { uri: filepath };
        } else {
          let { uri } = await FileSystem.downloadAsync(source.uri, filepath);
          source = { uri };
        }
      }
    } catch (e) {
      console.log(e);
    } finally {
      this.setState({ source });
    }
  }

  render() {
    if (this.state.source) {
      return <Image {...this.props} source={this.state.source} />;
    } else {
      let safeImageStyle = { ...StyleSheet.flatten(this.props.style) };
      delete safeImageStyle.tintColor;
      delete safeImageStyle.resizeMode;
      return <View style={safeImageStyle} />;
    }
  }
}

Thank you, your code helped me a lot.
Here is a version that was work for me, with updates for newer version of expo

import React from 'react';
import { Image, View, StyleSheet } from 'react-native';
import { Asset } from 'expo';
import * as FileSystem from 'expo-file-system';
import sha256 from 'crypto-js/sha256';

export default class CachedImage extends React.Component {
	constructor(props) {
		super(props);
		this.state = {
			source: null,
		};
	}

	async componentDidMount() {
		let { source } = this.props;

		try {
			if (typeof source === 'number') {
				await Asset.fromModule(source).downloadAsync();

			} else if (source && source.uri) {
				const parts = source.uri.split('.');
				const ext = parts[parts.length - 1];
				const name = sha256(source.uri);
				const filepath = `${FileSystem.cacheDirectory}${name}.${ext}'`;
				const { exists } = await FileSystem.getInfoAsync(filepath);
				if (exists) {
					source = { uri: filepath };
				} else {
					const { uri } = await FileSystem.downloadAsync(source.uri, filepath);
					source = { uri };
				}
			}
		} catch (e) {
			console.log(e);
		} finally {
			this.setState({ source });
		}
	}

	render() {
		const { source } = this.state;
		const { style } = this.props;

		if (source) {
			return <Image {...this.props} source={source} />;
		}

		const safeImageStyle = { ...StyleSheet.flatten(style) };
		delete safeImageStyle.tintColor;
		delete safeImageStyle.resizeMode;
		return <View style={safeImageStyle} />;
	}
}